FLASH GEOMETRY OF ALGEBRAIC CURVES

TRISTRAM DE PIRO

Abstract. In this paper, we develop the theory of flashes of an algebraic curve. We show that the theory is birationally invariant in a sense which we will make more precise below. We also show how the theory provides a foundation for the method of asymptotic degenerations, a particular class of degenerations of plane projective algebraic curves. In particular, we consider the geometrical technique in relation to the Severi problem of degenerating nodal curves to lines in general position, and correct Severi’s original proof of his conjecture.

1. Degenerating Nodal Curves

In this section, we return to the convention in [6] that a plane algebraic curve C is an irreducible subvariety of P^{2}, of dimension 1. By a node, we mean an ordinary double point, as in Definition 1.2. of [8]. By a plane nodal curve, we will mean a plane algebraic curve which has at most ordinary double points as singularities, as defined in [8]. In particular, this convention is different from the use of the term ”node” in [6], see also the explanation in [8]. We will also occasionally make use of the further assumption on C, see [6] and [7];

A generic point of C has character (1, 1) (†)

This is to exclude certain exceptional curves in non-zero characteristic. We will return to these exceptional cases in the final section.

An interesting geometric problem in the theory of nodal curves is to provide a general method of ”degenerating” a plane nodal curve C of degree m into a union of m lines, having transverse intersections. More specifically, if C is a plane nodal curve of degree m, one can consider degenerations, over an irreducible parameter space, which, for convenience of terminology, we denote by P^{1}, of the form {C_{t}}_{t∈P^{1}} with the following properties;

(i). The family {C_{t} : t ∈ P^{1}} consists of plane reduced algebraic curves.

(ii). C_{0} = C.

(iii). For any t ∈ P^{1}, all the singularities of C_{t} are nodes.

(iv). For any t ∈ (P^{1} \ ∞), C_{t} is a nodal curve (in particularly irreducible) and has the same number of singular points as C.

(v) C_{∞} is a union of m lines {l_{1}, …, l_{m}}. in general position, that is, the lines are distinct and, if m ≥ 3, there does not exist a triple (j_{1}, j_{2}, j_{3}), with j_{1} < j_{2} < j_{3} such that (l_{j_{1}} ∩ l_{j_{2}}) = (l_{j_{2}} ∩ l_{j_{3}}).

Severi referred to a plane nodal curve, having such a degeneration, as a ”nodal curve of the main stream”. He conjectured that any nodal curve is a nodal curve of the main stream. However, his proof of the result is erroneous.

We do not attempt to answer this conjecture fully in the following paper. However, it is hoped that the general theory of flash geometry, which we will develop here, can help to resolve the fallacious steps in Severi’s argument. We will return to this question in a later section.

The following lemma will be used repeatedly in arguments concerning nodal curves. The reader should look at [6] or [7] for relevant terminology.

Lemma 1.1. Nodal Curve Presentation

Let C be a plane nodal curve of degree m, satisfying (†). Then there exists an affine coordinate system (x, y) for P^{2}, such that, in this coordinate system;

(i). The line at ∞ cuts C transversely in m distinct non-singular points.

(ii). The tangent lines to C parallel to the y-axis all have 2-fold contact(contatto), and are based at non-singular points of C, which are in finite position.

(iii). Considering x as a rational function on C, in the terminology of [7], the weighted set G = (x = 0) consists of m distinct branches, each counted once, lying inside NonSing(C). Moreover, these branches are all in finite position, with base points distinct from the points of contact in (ii).

(iv). There is no line l, parallel to the y-axis, intersecting C in more than one point of the set consisting of its finitely many ordinary double points or the finitely many points named in (ii).

Proof. By the assumption (†) and Remarks 6.6 of [6], there exist only finitely many non-ordinary branches. Hence, there exist finitely many tangent lines {l_{γ_{1}}, …, l_{γ_{r}}}, based at {p_{1}, …, p_{r}}, (possibly with repetitions), such that;

I_{γ_{j}}(C, p_{j}, l_{γ_{j}}) ≥ 3, (for 1 ≤ j ≤ r)

Moreover, C has finitely many ordinary double points {q_{1}, …, q_{s}} as singularities. Let {l_{γ_{q_{1}}^{1}}, l_{γ_{q_{1}}^{2}}, …, l_{γ_{q^{1}_{s}}}, l_{γ_{q^{2}_{s}}}} be the 2s tangent lines, (possibly with repetitions), corresponding to these ordinary double points. Let {l_{q_{i}q_{j}} : 1 ≤ i < j ≤ s} be the finitely many lines, (possibly with repetitions), passing through at least 2 of these ordinary double points. Let {l_{1}, …, l_{t}} be the finitely many lines which are tangent to at least 2 distinct branches of C. That there exist finitely many such lines follows from duality arguments, the reader should look at Lemma 5.11 of [10]. Finally, let {l_{1}, …, l_{w}} be the finitely many lines, which have the property that, they pass through one of the ordinary double points {q_{1}, …, q_{s}}, and are also tangent to a branch, centred at a nonsingular point. That there are finitely many such lines follows from consideration of each of the finite number of pencils centred at {q_{1}, …, q_{s}}, and results from [6].

Now choose a point P not lying on C or any of the above defined lines. Let Σ = {l_{λ}^{P} }_{λ∈P^{1}} be the pencil defined by all lines passing through the point P. Then Σ defines a g_{m}^{1} on C without fixed branches. By Lemma 2.17 of [6], for generic λ, l_{λ}^{P} intersects C transversely in m distinct branches, based at non singular points of C. By construction, we also have that, if l_{P x} belongs to Σ and defines the tangent line l_{γ_{x}} to a branch γ_{x} based at x, then x ∈ NonSing(C), l_{P x} has 2-fold contact (contatto) with the branch γ_{x}, and l_{P x} cannot pass through either an ordinary double point or be tangent to another branch γ_{x′} of C, for x′ distinct from x. (∗). Moreover, if l_{P x} belongs to Σ, it can pass through at most one ordinary double point of C, (∗∗). Now choose a homography, sending the point [0 : 1 : 0] and the line Z′′ = 0, in the original coordinates [X′′ : Y′′ : Z′′] on P^{2}, to P and l_{λ}^{P}. Let [X′ : Y′ : Z′] be the new coordinate system defined by this homography. For the affine coordinate system (x′, y′), defined by x′ = \frac{X′}{Z′} and y′ = \frac{Y′}{Z′}, we have that the line at ∞ has the property (i). The lines parallel to the y′- axis correspond to the lines, excluding l_{λ}^{P}, in the pencil defined by Σ, in this new coordinate system. Hence, (ii) and (iv) follows immediately from (∗) and (∗∗). Now, considering x′ as a rational function on C, by the same argument as above, for generic μ ≠ ∞, the line defined by (x′ = μ), intersects C transversely in m distinct non-singular branches. Moreover, as P ([0 : 1 : 0] in this new coordinate system) does not lie on C, it follows that all these intersections lie in finite position, (∗ ∗ ∗). Now, let θ be the homography of P^{2} defined by the affine translation (x′, y′) ↦ (x′ − μ, y′) and let (x, y) be the new coordinates defined by this map. As P and the line at ∞ are fixed by θ, and, moreover, the tangent lines parallel to the y′-axis are translated to tangent lines parallel to the y-axis, conditions (i), (ii) and (iv) are preserved. By (∗∗∗), we then have that (iii) holds as well.

□

2. Newtonian Methods

We now discuss some geometric methods, originally developed by Isaac Newton, that constitute the main ideas behind the technique of flash geometry.

Definition 2.1. Asymptotes, Hyperbolic and Parabolic Branches

Let C be a plane irreducible algebraic curve of degree m, defined in the coordinate system (x, y), not equal to a line. We define a line l_{(a,b,t)}, given by the equation ax + by = t, to be an asymptote if;

(i). l_{(a,b,t)} passes through one of the intersections p ∈ C ∩ l_{∞}

(ii). l_{(a,b,t)} is tangent to at least one of the branches γ_{p} centred at p.

We define a branch γ_{p}, centred at one of the intersections p ∈ C ∩l_{∞}, to be an infinite branch, and a finite branch otherwise. We also define;

γ_{p} to be parabolic if the line l_{∞} is the tangent line to the branch. γ_{p} to be hyperbolic otherwise.

If C is equal to a line l, and this line does not coincide with the line l_{∞}, we define l to be its asymptote, and its one infinite branch γ_{∞} to be hyperbolic. If C is the line l_{∞}, we say that it has no asymptotes and all its infinite branches are parabolic.

If C is not irreducible, we consider its distinct irreducible factors {C_{1}, …, C_{k}}. The asymptotes of C are defined as the union of the asymptotes of each factor C_{j}, for 1 ≤ j ≤ k. Similarly, its branches are classified according to the definition above, for each irreducible factor.

As a simple consequence of this definition, we have;

Lemma 2.2. Let notation be as in Definition 2.1 and let C be any plane algebraic curve (possibly not irreducible) of degree m, having finite intersection with l_{∞}. Then C has r_{1} ≤ m parabolic branches and r_{2} ≤ m hyperbolic branches with r_{1} + r_{2} ≤ m. In particular there exist r_{3} ≤ r_{2} ≤ m distinct asymptotes to the curve C. If C is a plane nodal curve and the coordinate system is chosen as in Lemma 1.1, then there exist exactly m asymptotes to the curve C. Proof. Suppose first that C is irreducible. As we may assume that C has finite intersection with l_{∞}, by the Hyperspatial Bezout Theorem (Theorem 2.3 of [6]), we have that;

∑_{infinite branches γ_{p}} I_{γ_{p}}(C, l_{∞}) = m (∗)

It follows trivially that r_{1} + r_{2} ≤ m. If l_{(a,b,t)} is an asymptote, then it defines the tangent line to an infinite branch γ_{p}. As l_{(a,b,t)} is distinct from the line l_{∞}, such a branch γ_{p} must be hyperbolic. Any (hyperbolic) branch has a unique tangent line, hence distinct asymptotes must define tangent lines to distinct hyperbolic branches. Therefore, r_{3} ≤ r_{2} ≤ m. If C is not irreducible, let {C_{1}, …, C_{j}, …, C_{k}} be its distinct irreducible factors, let r_{1,j} be the number of parabolic branches of C_{j} and let r_{2,j} be the number of hyperbolic branches of C_{j}. If deg(C_{j}) = m_{j}, we clearly have that m_{1} + … + m_{j} + … + m_{k} ≤ m, (∗∗). Applying the previous considerations to each irreducible factor C_{j}, we have that r_{1,j} +r_{2,j} ≤ m_{j}. Applying the above Definition 2.1 and (∗∗), we have that r_{1} + r_{2} ≤ m. Again, by a similar argument to the above, we have that r_{3} ≤ r_{2} ≤ m. For the final part of the lemma, by (i) of Lemma 1.1 and (∗), we have that there exist m distinct infinite hyperbolic branches. The tangent lines of these hyperbolic branches must be distinct, otherwise at least one would coincide with the line l_{∞}, hence there exist exactly m asymptotes as required. □

Remarks 2.3. An elegant method of determining the real asymptotes to a given plane real irreducible algebraic curve is given in [25]. I hope the author will not mind me essentially restating his results in the context of algebraically closed fields.

Theorem. (adapted almost verbatim from Nunemacher)

Let C be any any plane algebraic curve and let C be defined in the coordinate system (x, y) by;

P (x, y) = ∑_{k=0}^{m} P_{k}(x, y)

where P_{k}(x, y) is a homogeneous polynomial of degree k. Suppose that (ax + by) is a factor of the top degree form P_{m}(x, y) of multiplicity r. Let s ≤ r denote the largest integer with the property that there exist polynomials Q_{j}(x, y) for m−s+1 ≤ j ≤ m satisfying the conditions;

P_{m}(x, y) = (ax + by)^{s}Q_{m}(x, y)

P_{m−1}(x, y) = (ax + by)^{s−1}Q_{m−1}(x, y)

. . . . . .

P_{m−s+1}(x, y) = (ax + by)Q_{m−s+1}(x, y) (∗)

Then associated with the factor (ax+by) is a set of at most s asymptotes ax + by = t_{0}, where t_{0} is a root of the equation;

t^{s}Q_{m}(b, −a)+t^{s−1}Q_{m−1}(b, −a)+…+tQ_{m−s+1}(x, y)+P_{m−s}(b, −a) = 0

All the asymptotes to the curve C in the coordinate system (x, y) arise in this way as (ax + by) ranges over the linear factors of P_{m}(x, y)

Corollary. (adapted almost verbatim from Nunemacher)

With the notation and hypotheses of the above theorem, if (ax + by) is a simple factor of P_{m}(x, y), then associated with this factor is the single asymptote to C, defined by the equation;

(ax + by)Q_{m}(b, −a) + P_{m−1}(b, −a) = 0

The proof of the Theorem and its Corollary follow easily from the paper [25]. We have strengthened the hypotheses of the Theorem to include any plane algebraic curve, the corresponding Theorem in [25] excludes the degenerate case that C contains a line as one of its irreducible factors. However, given Definition 2.1, which includes this case, one can check that his proof is unaffected in this greater degree of generality.

In order to give an illustration of the theorem and its corollary, consider the following curves;

(i). The hyperbola defined by xy = λ.

In this case, we have that P_{2}(x, y) = xy, whose simple linear factors are x and y, and P_{1}(x, y) = 0. By the corollary, the asymptotes are given by;

x.y(0, −1) + 0(0, −1) = 0 that is x = 0

y.x(1, 0) + 0(1, 0) = 0 that is y = 0

(Observe that this gives the correct result even in the degenerate case when λ = 0)

(ii). The parabola defined by x^{2} − y = 0.

In this case, we have that P_{2}(x, y) = x^{2}, which has x as a linear factor of multiplicity 2, and P_{1}(x, y) = −y. We have that x does not divide P_{1}(x, y), hence, by the theorem, the asymptotes are given by x = t_{0}, where t_{0} is a root of;

t.x(0, −1) − y(0, −1) = 0 that is 1 = 0

As this has no solutions, we conclude that the parabola has no asymptotes.

(iii). The curve defined by;

8 TRISTRAM DE PIRO

xy^{2} + ey = a^{2}x^{3} + bx^{2} + cx + d, see (1) of Remarks 2.4. Case 1. a ≠ 0. In this case, P_{3}(x, y) = xy^{2}−a^{2}x^{3}, which, assuming char(L) ≠ 2, has

three distinct linear factors {x, (y+ax), (y−ax)}, and P_{2}(x, y) = −bx^{2}. By the corollary, the asymptotes are given by;

x.(y^{2} − a^{2}x^{2})(0, −1) + (−bx^{2})(0, −1) = 0 that is x = 0 (y − ax)(xy + ax^{2})(1, a) + (−bx^{2})(1, a) that is y − ax = \frac{b}{2a} (y + ax)(xy − ax^{2})(1, −a) + (−bx^{2})(1, −a) that is y + ax = \frac{−b}{2a} The curve therefore has three distinct asymptotes, see (2) of Remarks

2.4. If b = 0, these asymptotes intersect at the origin (0, 0), see (3) of Remarks 2.5. If b ≠ 0, these asymptotes intersect in a triangle with vertices centred at \{(0, \frac{b}{2a}), (0, \frac{−b}{2a}), (\frac{−b}{2a^{2}} , 0)\}, see (4) of Remarks 2.4.

Case 2. a = 0, b ≠ 0. In this case, P_{3}(x, y) = xy^{2}, which has x as a simple factor and y as

a linear factor of multiplicity 2, again P_{2}(x, y) = −bx^{2}. By the corollary, one asymptote is given by;

x.y^{2}(0, −1) + (−bx^{2})(0, −1) = 0 that is x = 0. As y does not divide P_{2}(x, y), the theorem gives the other asymptotes

as y = t_{0}, where t_{0} is a root of; t.(xy)(1, 0) + (−bx^{2})(1, 0) = 0 that is b = 0 As this has no solutions, we conclude there are no further asymp-

totes, see (5) of Remarks 2.4.

Case 3. a = 0, b = 0. By the same reasoning as Case 2, we obtain x = 0 as an asymptote.

In this case, however, y divides P_{2}(x, y) = 0, so we have to consider P_{1}(x, y) = ey−cx. By the theorem, the other asymptotes are then given by y = t_{0}, where t_{0} is a root of;

FLASH GEOMETRY OF ALGEBRAIC CURVES 9

t^{2}.x(1, 0) + t.0(1, 0) + (ey − cx)(1, 0), that is t^{2} − c = 0 In the case when c = 0, we obtain one further asymptote y = 0, oth-

erwise, assuming char(L) ≠ 2, we obtain 2 further asymptotes y = c^{\frac{1}{2}} and y = −c^{\frac{1}{2}} , see (6) of Remarks 2.4.

(iv). The curve defined by; xy = ax^{3} + bx^{2} + cx + d = 0, see (7) of Remarks 2.4. We have that P_{3}(x, y) = ax^{3}, which has x as a linear factor of mul-

tiplicity 3, P_{2}(x, y) = bx^{2} − xy. By the theorem, the asymptotes are given by x = t_{0}, where t_{0} is a root of;

t^{2}.x(0, −1) + t.(bx − y)(0, −1) + cx(0, −1) = 0 that is t = 0 Hence, x = 0 is the only asymptote to this curve. (v). The curve defined by; y^{2} = ax^{3} + bx^{2} + cx + d, with a ≠ 0, see (8) of Remarks 2.4. We have that P_{3}(x, y) = ax^{3}, which has x as a linear factor of multi-

plicity 3, P_{2}(x, y) = bx^{2} −y^{2}. By the theorem, the asymptotes are given by x = t_{0}, where t_{0} is a root of;

t.x^{2}(0, −1) + (bx^{2} − y^{2})(0, −1) = 0, that is −1 = 0 This has no solutions, hence the curve has no asymptotes. (vi). The curve defined by; y = ax^{3} + bx^{2} + cx + d, with a ≠ 0, see (9) of Remarks 2.4 We have that P_{3}(x, y) = ax^{3}, which has x as a linear factor of mul-

tiplicity 3, P_{2}(x, y) = bx^{2}, and P_{1}(x, y) = cx − y. By the theorem, the asymptotes are given by x = t^{0}, where t_{0} is a root of;

t^{2}.x(0, −1) + t.bx(0, −1) + (cx − y)(0, −1) = 0, that is 1 = 0

This has no solutions, hence the curve has no asymptotes.

Remarks 2.4. Isaac Newton gives the first effective method of determining the asymptotes of a real algebraic curve C in [22]. The technique is part of a more general construction called ”Newton’s parallelogram”. An excellent account of this construction is given in [27]. Moreover, Newton makes a systematic use of the method of asymptotes in his classification of real cubic curves, see [23] and [27]. His analysis of such curves in [23] begins with the observation;

”All lines of the 1st, 3rd, 5th, 7th, or odd orders, have at least two infinite branches extending in opposite directions”

In the case of a real cubic curve C, this translates to the fact that the line l_{∞} intersects C in at least 1 point, counted with multiplicity (appropriately defined). The observation is easily seen to be true from the fact that any real polynomial p ∈ ℝ[y] of degree 3 has at least 1 real solution and consideration the limiting behavior of such a solution as x → ∞ or x → −∞ in the defining equation P (x, y) of C. Excluding the degenerate case when C has a parabolic branch, that is the line l_{∞} intersects C in a point with multiplicity 2, we obtain a hyperbolic branch and at least 1 asymptote to the curve C. In this case, by taking such an asymptote to be the coordinate axis x = 0, the general equation of C reduces to the form;

xy^{2} + (bx^{2} + cx + d)y = cx^{3} + dx^{2} + ex + f (∗)

Such a quadratic equation can be analysed using the simple method of completing the square. This construction forms the basis for Newton’s division of real cubics into four Cases. These Cases are then further analysed according to the behaviour of their asymptotes and the existence of a diameter (see [23] and [27] for a precise definition.) This leads to Newton’s division of real cubics into a total of 72 species. The curves considered as examples in the previous remark constitute a number of these species;

(1). Newton refers to this equation as The Case I cubic, see [23].

(2). Newton refers to such a curve, in [23], as a redundant or triple hyperbola (if a^{2} > 0), and as a defective hyperbola (if a^{2} < 0).

(3). The triple hyperbolas with this property are considered as Species 24-32 in Newton’s classification of cubics, see [23].

(4). The triple hyperbolas with this property are considered as Species 1-23 in Newton’s classification of cubics (and are subdivided further according to the number of diameters). The defective hyperbolas are considered as Species 33-45 in Newton’s classification, see [23].

(5). Newton refers to such curves as parabolic hyperbolas, they account for Species 46-56 of his classification, see [23].

(6). Newton refers to such curves as the hyperbolisms of the hyperbola ,(c > 0), ellipse, (c < 0), or the parabola, (c = 0), they account for species 57-60, species 61-63 and species 64-65 respectively, see [23].

(7). Newton calls this equation The Case II cubic, see [23], also known as Newton’s trident.

(8). Newton calls this equation The Case III cubic, see [23], referred to there as the Divergent Parabola.

(9). Newton calls this equation The Case IV cubic, see [23], referred to there as the Cubic Parabola.

Perhaps the most important ingredient in Newton’s analysis of cubics is the use that he makes of asymptotes as a means of representing cubic curves. A purely algebraic proof of his result would be inconceivable without the supporting intuition of such a representation. The reader is strongly encouraged to look at the illustrations that Newton originally gave of each species in his classification, see [27]. The fact that Newton referred to a cubic curve (curve of degree m) as a ”line of the third order” (a ”line of the m’th order”), clearly shows that he thought of such curves in terms of a system of lines, (†), (^{1}). I discuss this idea in greater detail in my forthcoming book ”Christian Geometry”, where I consider the aesthetics ideas underlying Newton’s work. The principle (†) is also illustrated by the problem of degenerating curves of degree m to a union of m lines, which we discussed in the first section.

1In the following section, this viewpoint will become clearer when we introduce the notion of flashes, as a means of representing an algebraic curve.

Using the coordinate system of Lemma 1.1, we can find the following decomposition of C;

Lemma 2.5. Newton’s Theorem

Let hypotheses be as in Lemma 1.1, and let F (x, y) define C in the coordinate system (x, y) given by Lemma 1.1. Then we can find algebraic power series {η_{1}(x), …, η_{m}(x)} in L[[x]], with {η_{1}(0), …, η_{m}(0)} distinct, such that;

F (x, y) = (y − η_{1}(x)) … (y − η_{m}(x))

as an identity in the ring L[[x]][y].

Proof. Let {a_{1}, …, a_{m}} enumerate the m distinct solutions to F (0, y) = 0. Making the linear change of coordinates y′ = y − a_{j}, the equation for C in this system is given by;

F_{j}(x, y′) = F (x, y′ + a_{j}) = 0 (†)

By construction, we have that F_{j}(0, 0) = 0 and, moreover, by (iii) of Lemma 1.1, \frac{∂F_{j}}{∂y} (0, 0) ≠ 0. It follows, applying the implicit function theorem, that we can find an algebraic power series δ_{j}(x) ∈ L[[x]], with δ_{j}(0) = 0, such that F_{j}(x, δ_{j}(x)) = 0, (∗), see especially the following remark. Now let η_{j}(x) = a_{j} + δ_{j}(x). By (†), it follows immediately that F (x, η_{j}(x)) = 0, for 1 ≤ j ≤ m. Hence, we have that;

(y − η_{j}(x))|F (x, y), for 1 ≤ j ≤ m, in L((x))[y]

using the fact that the division property holds in the polynomial ring L((x))[y], as L((x)) is a field. We clearly have that each (y − η_{j}(x)) is irreducible in L((x))[y], hence prime. Moreover, as the power series {η_{1}(x), …, η_{m}(x)} are distinct, the factors (y − η_{j}(x)) are coprime in L((x))[y], for 1 ≤ j ≤ m. It, therefore, follows that;

∏_{1≤j≤m}(y − η_{j}(x))|F (x, y) in L((x))[y] (∗∗)

Now, using the fact that both sides of the relation have degree m, we can find g(x) ∈ L((x)) such that;

F (x, y) = g(x)(y − η_{1}(x)) … (y − η_{m}(x)) in L((x))[y]

As F (0, y) = 0 has m distinct solutions in L, we can write F in the form;

F (x, y) = y^{m} + p_{1}(x)y^{m−1} + … + p_{j}(x)y^{m−j} + … + p_{m}(x)

with p_{j} ∈ L[x] and deg(p_{j}) = j. It follows that we can take g(x) = 1 in (∗∗), so that the identity holds in L[[x]][y]. This shows the lemma.

□

Remarks 2.6. The existence of power series solutions to polynomial equations G(x, y) = 0, satisfying (∗) of the previous lemma, was first shown by Isaac Newton. He gives the following method of constructing a solution by successive approximations, in ”The Praxis of Resolution”, Paragraph 36 of [24];

The assumption that G(0, 0) = 0 and \frac{∂G}{∂y}(0, 0) ≠ 0 allows us to write;

G(x, y) = p_{m}(x)y^{m} + … + p_{1}(x)y + p_{0}(x) (∗)

with p_{0}(0) = 0 and p_{1}(0) ≠ 0. For (x, y) ”small”, Newton observes that it, therefore, makes sense to take as a first approximate solution;

y_{0} = \frac{−λ_{0}x^{i_{0}}}{p_{1}(0)}

where λ_{0}x^{i_{0}}, (i_{0} ≥ 1), is the first term in the expression for p_{0}(x). Now, Newton makes the substitution y = (y′ + y_{0}) in (∗), this results in a further polynomial equation of the same form;

q_{m}(x)y^{′m} + … + q_{1}(x)y′ + q_{0}(x) = 0 (∗∗)

By a straightforward algebraic calculation, using the fact that i_{0} ≥ 1, one checks that q_{1}(0) ≠ 0 and ord(q_{0}(x)) > ord(p_{0}(x)). Hence, one can take as the second quote;

y_{1} = \frac{−λ_{1}x^{i_{1}}}{q_{1}(0)}

where λ_{1}x^{i_{1}}, (i_{1} > i_{0}), is the first term in the expression for q_{0}(x). Continuing in this way, one obtains a sequence of approximate solutions;

s_{n}(x) = y_{0}(x) + y_{1}(x) + … + y_{n}(x), for n ≥ 0

Either this process terminates after a finite number of approximations, giving a polynomial solution to (∗), or one obtains an infinite power series;

s(x) = ∑_{i≥0} y_{i}(x)

A straightforward algebraic calculation shows that;

ord(G(x, s_{n+1}(x))) ≥ ord(G(x, s_{n}(x))) + 1

Hence, by elementary completeness arguments for the power series ring L[[x]], we are guaranteed that G(x, s(x)) = 0.

This method of constructing a solution to the polynomial equation G(x, y) = 0 is closely related to the possibly more familiar Newton-Raphson method. Namely, one considers the function;

G : L[x] → L[x], G(y) = p_{m}(x)y^{m} + … + p_{1}(x)y + p_{0}(x) (†)

Having obtained a first approximation y_{0} = s_{0} to the equation G(y) = 0, the Newton-Raphson method gives the further approximation;

s_{1} = y_{0} − \frac{G(y_{0})}{G′(y_{0})} = y_{0}(x) − \frac{q_{0}(x)}{q_{1}(x)}

where q_{0}(x) and q_{1}(x) are obtained from the transformed polynomial (∗∗) above. By a similar argument to the above, replacing the successive approximations −\frac{λ_{1}x^{i_{1}}}{q_{1}(0)} by −\frac{q_{0}(x)}{q_{1}(x)} and noting that;

\frac{q_{0}(x)}{q_{1}(x)} = \frac{λ_{1}x^{i_{1}}}{q_{1}(0)} u(x) for a unit u(x) ∈ L[[x]]

one is similarly guaranteed that this method also yields the same power series solution s(x) to the equation G(y) = 0 in (†). The Newton-Raphson method is usually applied to functions of a real variable, rather than the rings L[x] or L[[x]]. However, that Newton intended his method of finding power series solutions to polynomial equations of the form G(x, y) = 0, (”species”), to be a partial generalisation of this method is borne out by his consideration of the case of ”affected equations”, at the beginning of [24]. An affected equation is just the case where the indeterminate x is replaced by an explicit numerical value.

Newton also gives a general method for finding power series solutions to polynomial equations of the form G(x, y) = 0, without the simplifying assumption that \frac{∂G}{∂y}(0, 0) ≠ 0. This is done by the introduction of ”Newton’s parallelogram”, in Paragraph 29 of [24], and fractional exponents x^{\frac{1}{t}}, for t ≥ 1. The parallelogram method is further explained in [22]. This leads to a general method of finding n solutions y_{j}(x) ∈ L((x^{\frac{1}{t}})), for 1 ≤ j ≤ n, of a polynomial equation G(x, y) = 0 of degree n, see [1] for a more precise statement of ”Newton’s Theorem”. However, it seems clear from Newton’s highly geometric approach, that he intended the theorem to provide a method of understanding the global structure of an algebraic curve, rather than as a means of analysing the nature of local singularities. This view is supported by consideration of his work on the classification of plane cubics, in [23]. Here, his power series method is used to find the asymptotes of such curves, which he refers to as ”infinite branches”, and branches at finite distances (those crossing the axis x = 0), also referred to in the literature as ”satellite branches”. Much of the work in [23] is concerned with ”filling in” the rest of the curve, from this initial information. This is also the approach taken by a number of Newton’s followers in England, see, for example, the papers [26],[21] and the commentary on [23], in [27]. As one can always choose the ”satellite branches” to be non-singular, see Lemma 1.1, one can obtain a solution to ”Newton’s Theorem” without introducing ”Puiseux Series”, as, indeed, is done in the previous Lemma 2.5. It is extremely unclear how to give a geometric interpretation of ”Puiseux Series”. It seems, therefore, to be a rather unfortunate historical accident that these were introduced, as a response to Newton’s work, in order to analyse algebraic curves. The proofs that we give in the following section, which develop flash geometry, closely follows Newton’s original approach.

I also discuss the general aesthetics behind Newton’s representation of algebraic curves in ”Christian Geometry” (in preparation). As already mentioned, the reader would benefit greatly from looking at Newton’s original ”sketches” of cubic curves in [23].

3. Flash Geometry of Nodal Curves

Definition 3.1. We will denote the coordinates of P^{2} by {X, Y, W } or {X, Z, W }, the coordinates of P^{3} by {X, Y, Z, W } and the points {[0 : 0 : 1 : 0], [0 : 1 : 0 : 0]} by {Q_{1}, Q_{2}}. We let {pr_{1}, pr_{2}} be the canonical conic projection morphisms defined by;

pr_{1} : (P^{3} \ Q_{1}) → P_{1}^{2}; pr_{1}([X : Y : Z : W]) = [X : Y : 0 : W] pr_{2} : (P^{3} \ Q_{2}) → P_{2}^{2}; pr_{2}([X : Y : Z : W]) = [X : 0 : Z : W]

We let A_{1}^{2} ⊂ P_{1}^{2} be the open subset defined by the coordinate system (x, y), where x = \frac{X}{W} and y = \frac{Y}{W}, and l_{1,∞} ⊂ P_{1}^{2} be the line at infinity defined by (W = 0) ∩ P_{1}^{2}. Similarly, we let A_{2}^{2} ⊂ P_{2}^{2} be the open subset defined by the coordinate system (x, z), where x = \frac{X}{W} and z = \frac{Z}{W}, and l_{2,∞} ⊂ P_{2}^{2} the line at infinity defined by (W = 0) ∩ P_{2}^{2}. We let A^{3} ⊂ P^{3} be the open subset defined by the coordinate system (x, y, z), where x = \frac{X}{W}, y = \frac{Y}{W} and z = \frac{Z}{W}. The restriction of the projection morphisms {pr_{1}, pr_{2}} are then given in affine coordinates by;

pr_{1} : A^{3} → A_{1}^{2}; pr_{1}(x, y, z) = (x, y)

pr_{2} : A^{3} → A_{2}^{2}; pr_{2}(x, y, z) = (x, z)

We will also refer to the projected points {pr_{2}(Q_{1}), pr_{1}(Q_{2})} as {Q_{1}, Q_{2}}, and the restriction of the projections {pr_{1}, pr_{2}} to {A_{2}^{2}, A_{1}^{2}} by pr.

We observe the following straightforward property;

Lemma 3.2. There does not exist a pair {a, b}, distinct from {Q_{1}, Q_{2}}, with \overline{pr_{1}^{−1}(a)} = \overline{pr_{2}^{−1}(b)}.

Proof. Observe that a fibre of pr_{1} consists of a line passing through Q_{1}, with the point Q_{1} removed. Similar considerations apply to pr_{2}, (†). Suppose a pair, as in the lemma, existed. If {a, b} are distinct, then, by (†), we would clearly have that \overline{pr_{1}^{−1}(a)} = \overline{pr_{2}^{−1}(b)} = l_{ab} = l_{Q_{1}Q_{2}}. As the image of pr_{1}∘pr_{2} is P_{1}^{2}∩P_{2}^{2}, this implies that l_{ab} = P_{1}^{2}∩P_{2}^{2}. As P_{1}^{2}∩P_{2}^{2} is fixed both by pr_{1} and pr_{2}, (∗), it follows that a = pr_{2}(a) = pr_{1}(b) = b. Hence, we have that pr_{1}^{−1}(a) = pr_{2}^{−1}(a) = P_{1}^{2} ∩ P_{2}^{2}, which clearly contradicts (∗).

□

We now show the following;

Lemma 3.3. Let hypotheses be as in Lemma 2.5 and notation as in Definition 3.1, then we can find an irreducible plane projective curve C′ ⊂ P^{2}, relative to the coordinate system (x, z), such that;

(i). There exists an open V′ ⊂ C′, with (0, 0) ∈ V′, such that;

FLASH GEOMETRY OF ALGEBRAIC CURVES 17

pr : (V′, (00)) → (A^{1}, 0), (∗) is an etale morphism. (ii). {δ_{1}(x), …, δ_{m}(x)} ⊂ R(V′), for a given embedding of R(V′) in

L[[x]] ∩ L(x)^{alg}.

(iii). The function field L(V′) is generated over L(x) by the power series {δ_{1}(x), …, δ_{m}(x)}.

(iv). The cover defined by (∗) is Galois, with the extension L(V′)/L(x) equal to the Galois closure of the extension L(C)/L(x).

Proof. From the previous construction of Lemma 2.5, the power series {δ_{1}(x), …, δ_{m}(x)} belong to L[[x]] ∩ L(x)^{alg}. By Theorem 1.3 and subsequent remarks at the beginning of Section 3 of [5], we have that L[[x]] ∩ L(x)^{alg} is isomorphic to the local ring for the etale topology O_{(A^{1},0)}^{et}.

By the definition of this local ring, we can find irreducible varieties; {(U_{1}, (0)_{1}^{lift}), …, (U_{i}, (0)_{i}^{lift}), …, (U_{m}, (0)_{m}^{lift})} and etale morphisms; ϕ_{i} : (U_{i}, (0)_{i}^{lift}) → (A^{1}, 0), for 1 ≤ i ≤ m with δ_{i}(x) ∈ R(U_{i}). It follows, from algebraic considerations, see, for

example, [17], that the fibre product; (U_{1...m}, (0)^{lift}) = (U_{1} ×_{A^{1}} × … ×_{A^{1}} U_{m}, (0)_{1}^{lift} ×_{(0)} … ×_{(0)} (0)_{m}^{lift}) is irreducible, nonsingular, and defines etale covers; ψ_{1...m} : (U_{1...m}, (0^{lift})) → (A^{1}, 0) ψ_{i} : (U_{1...m}, (0^{lift})) → (U_{i}, (0)_{i}^{lift}), for 1 ≤ i ≤ m such that ϕ_{i} ∘ ψ_{i} = ψ_{1...m}. By the definition of the etale local ring,

in terms of maps and equivalence relations, we can assume that all the power series {δ_{1}(x), …, δ_{m}(x)} are represented in the coordinate ring R(U_{1...m}).

18 TRISTRAM DE PIRO

The dominant morphism ψ_{1...m} induces an inclusion; L(x) ⊂ L(U_{1...m}) and a corresponding factorisation; L(x) ⊂_{i_{1}} L(x, δ_{1}(x), …, δ_{m}(x)) ⊂_{i_{2}} L(U_{1...m}),(^{2}) Let C′′ be an irreducible nonsingular model of L(x, δ_{1}(x), …, δ_{m}(x)),

with corresponding dominant morphisms; α_{2} : U_{1...m} → C′′, α_{1} : C′′ → A^{1} corresponding to the inclusions {i_{1}, i_{2}}, such that α_{1} ∘ α_{2} = ψ_{1...m}.

Let (0)^{lift} ∈ C′′ also denote the image α_{2}((0)^{lift}), for (0)^{lift} ∈ U_{1...m}. As the composition α_{1} ∘ α_{2} is etale, it follows, using an elementary Zariski structure argument, that the morphism α_{1} is Zariski unramified at (0)^{lift} ∈ C′′. By nonsingularity of C′′, Footnote 2 and Theorems (6.10,6.11) of [7], there exists an open subset W ⊂ C′′, containing (0)^{lift}, such that the restriction;

α_{1} : (W, (0)^{lift}) → (A^{1}, 0) is etale. We now use the local presentation lemma for etale morphisms, given

in Fact 1.5 of [5], that is we can find an open subset U′ ⊂ A^{1}, containing (0), and a monic polynomial F (z) ∈ R(U′)[z], such that α_{1} can be presented for an open subset V′ ⊂ W ⊂ C′′ in the form;

Spec((\frac{R(U′)[z]}{F (z)})_{d}) → Spec(R(U′)) (∗∗) with F′(z) invertible in (\frac{R(U′)[z]}{F (z)})_{d}. Now let p(x) ∈ L[x] be a polyno-

mial, with p(0) ≠ 0, such that R(U′) = L[x]_{p}. Then we can write the equation for F (z) in the form;

F (z) = z^{n} + \frac{c_{1}(x)}{p^{r_{1}}(x)}z^{n−1} + … + \frac{c_{n−1}(x)}{p^{r_{n−1}}(x)}z + \frac{c_{n}(x)}{p^{rn}(x)} = 0

2When char(L) ≠ 0, using the fact that the roots η_{1}(x), …, η_{m}(x), found in Lemma 2.5, are distinct, it is straightforward to check that L(x, δ_{1}(x), …, δ_{m}(x))

is a seperable extension of L(x).

for some n ≥ 1 and 0 ≤ r_{j} ≤ r, with c_{j}(x) coprime to p(x) for 1 ≤ j ≤ n. Clearing coefficients and using the fact that V′ is irreducible, the polynomial;

G(x, z) = p^{r}(x)z^{n} + c_{1}(x)p^{r−r_{1}}(x)z^{n−1} + … + c_{n}(x)p^{r−r_{n}}(x)

defines an irreducible curve C′ in A_{2}^{2}, whose restriction to V′ = (pr_{1}^{−1}(U′)∩(G = 0)∩(d ≠ 0)) corresponds to the cover defined in (∗∗). By making a vertical translation of the curve, we can assume that G(0, 0) = 0 and (0, 0) corresponds to the point (0)^{lift} of V′ ⊂ W ⊂ C′′. We finally check the properties (i) to (iv) for (V′, C′, (0, 0)).

(i). Follows immediately from the presentation (∗∗).

(ii). As there exists a birational map between (V′, C′′) and (V′, C′) over (A^{1}, 0), we can assume that the power series {δ_{1}(x), …, δ_{m}(x)} ⊂ R(V′) ⊂ L(C′). That δ_{1}(0, 0) = … = δ_{m}(0, 0) = 0, follows from the definition of the maximal ideal 𝔪_{(A^{1},0)}^{et} ⊂ O_{(A^{1},0)}^{et}.

(iii). Follows from the construction of C′′ and the fact that {C′, C′′} are birational.

(iv). Follows from (iii) and the fact that L(x, δ_{1}(x), …, δ_{m}(x)) = L(x, η_{1}(x), …, η_{m}(x)) is a splitting field for the polynomial F (x, y), given in Lemma 2.5.

□

The existence of a curve C′ ⊂ P^{2}, satisfying the conditions of the lemma, is far from unique. More precisely, we have the following;

Lemma 3.4. Let C_{1} satisfy the conditions of Lemma 3.3 and let;

g(x, z) = \frac{a(x)z+b(x)}{c(x)z+d(x)}

with {a(x), b(x), c(x), d(x)} polynomials in L[x], such that b(0) = 0 and a(0)d(0) ≠ 0

Then g(x, z) determines a birational,(^{3}), morphism θ_{g} : C_{1} ↭ C_{2}, with C_{2} satisfying the conditions of Lemma 3.3.

3Unless otherwise stated, when speaking of a birational morphism θ between curves C_{1} and C_{2}, satisfying the conditions of Lemma 3.3, we will always assume

Proof. As a(0)d(0) ≠ b(0)c(0), there exists an open subset U ⊂ A^{1}, containing 0, such that, for all x ∈ U, g(x, z) = \frac{a(x)z+b(x)}{c(x)z+d(x)} defines an invertible Mobius transformation. Let;

V_{1} = {(x, z) ∈ U × A^{1} : c(x)z + d(x) ≠ 0}

V_{2} = {(x, z) ∈ U × A^{1} : c(x)z − a(x) ≠ 0}

Then the morphism;

θ_{g} : V_{1} → V_{2}; (x, z) ↦ (x, g(x, z))

has an inverse provided by the transformation;

θ_{g}^{−1} : V_{2} → V_{1}; (x, z) ↦ (x, g^{−1}(x, z))

where g^{−1} is the inverse Mobius transformation to g. Moreover, we have that pr_{1} ∘ θ_{g} = Id_{(U\c(x)=0)}.

The image of the irreducible open subset (C_{1} ∩ V_{1}) ⊂ C_{1} is constructible, hence, we can form its projective closure C_{2}. It follows immediately, that θ_{g} defines an isomorphism between the open subsets (C_{1} ∩ V_{1}) and (C_{2} ∩ V_{2}), (†), therefore, defines a birational map θ_{g} : C_{1} ↭ C_{2}. We check conditions (i) to (iv) for the curve C_{2}.

(i). We have that (0, 0) ∈ (V_{1} ∩ V_{2}) and g(0, 0) = 0. By (†), (0, 0) defines a nonsingular point of the curve C_{2}. In order to verify that pr_{1} defines an etale morphism in an open neighborhood of (0, 0) ∈ C_{2}, it is then sufficient to verify that the morphism is Zariski unramified at (0, 0), by Theorems (6.10,6.11) of [7]. Suppose not, then we can find ϵ ∈ (V_{0} ∩ A^{1}), and distinct {ϵ_{1}, ϵ_{2}} ⊂ V_{0} such that {(ϵ, ϵ_{1}), (ϵ, ϵ_{2})} ⊂ V_{(0,0)} ∩ C_{2} ∩ V_{1} ∩ V_{2}. It follows immediately that {(ϵ, g^{−1}(ϵ, ϵ_{1})), (ϵ, g^{−1}(ϵ, ϵ_{2}))} ⊂ V_{(0,0)} ∩C_{1} ∩V_{1} ∩V_{2}. Moreover, the pair {(ϵ, g^{−1}(ϵ, ϵ_{1})), (ϵ, g^{−1}(ϵ, ϵ_{2}))} is distinct, by birationality. This contradicts the fact that pr_{1} defines an etale morphism in an open neighborhood of (0, 0) ∈ C_{1}.

that θ is an isomorphism in the etale topology with (0, 0) as labelled points, and (pr ∘ θ) = pr. Obvious examples of birational morphisms which do not satisfy this requirement are non-identity elements of the automorphism group (L(C_{1})/L(x)).

It is not immediately obvious that, for any two curves C_{1} and C_{2}, satisfying the conditions of Lemma 3.3, there does exist a birational morphism between C_{1} and C_{2}, with the above extra requirement. However, this will be shown later in the paper.

The remaining conditions (ii), (iii), (iv) are straightforward to check, using the fact that;

θ_{g} : (C_{1} ∩ V_{1}, (0, 0)) → (C_{2} ∩ V_{2}, (0, 0))

is an isomorphism, with the property that (pr_{1} ∘ θ_{g}) = pr_{1}.

□

The verticalised, polar geometry of such birational transformations, suggests that we should give special consideration to the effect on branches, centred at the point Q_{1}, determined by the coordinate system (x, z);

Lemma 3.5. Effect on hyperbolic branches at Q_{1}

Let C_{1}, C_{2} and θ_{g} satisfy the conditions of the previous lemma and suppose that γ is a hyperbolic branch of C_{1}, centred at Q_{1}, with tangent line x = α. Then, the corresponding branch θ_{g}(γ) of C_{2}, determined by the birational map θ_{g}, is centred at Q_{1}, if c(α) = 0 and a(α) ≠ 0, and at (α, \frac{a(α)}{c(α)}), if c(α) ≠ 0.

Proof. As x = α is tangent to the branch γ, and not equal to l_{∞}, by results of [6], given an infinitesimal ϵ ∈ V_{0}, there exists z(ϵ) such that;

(α + ϵ, z(ϵ)) ∈ (V_{Q_{1}} ∩ C_{1} ∩ γ)

By the explicit construction of specialisations, given in [3], we may assume that z(ϵ) = \frac{u(ϵ)}{ϵ^{i}}, for a unit u(x) ∈ L[[x^{\frac{1}{r}}]], (r ≥ 1), and i > 0, i ∈ Q, (^{4})

If θ_{g}(γ) is centred at P, then, using results of [6];

4The use of Puiseux series, here, is necessary, due to the existence of blue branches (Definition 3.11). We will find a more effective method of understanding the geometry of such branches, later in the paper.

22 TRISTRAM DE PIRO

θ_{g}((α + ϵ, z(ϵ))) ∈ (V_{P} ∩ C_{2} ∩ θ_{g}(γ)) We explicitly determine P by specialisation in the following cases; Case 1. c(α) = 0 and a(α) ≠ 0 We have that; g(α + ϵ, z(ϵ)) = \frac{a(α+ϵ)z(ϵ)+b(α+ϵ)}{c(α+ϵ)z(ϵ)+d(α+ϵ)} = \frac{a(α+ϵ)u(ϵ)+b(α+ϵ)ϵ^{i}}{c(α+ϵ)u(ϵ)+d(α+ϵ)ϵ^{i}} hence, the projective coordinates of θ_{g}(ϵ, z(ϵ)), see Definition 3.1, are

given by; X = (α + ϵ)(c(α + ϵ)u(ϵ) + d(α + ϵ)ϵ^{i}) Z = a(α + ϵ)u(ϵ) + b(α + ϵ)ϵ^{i} W = c(α + ϵ)u(ϵ) + d(α + ϵ)ϵ^{i} Using the fact that c(α) = 0 and a(α) ≠ 0, the specialised coordi-

nates are given by; X = 0, Z = 1, W = 0 hence, the result follows. Case 2. c(α) ≠ 0. Writing z(ϵ) = \frac{1}{t(ϵ)}, where t(ϵ) = ϵ^{i}u(ϵ), we have that; θ_{g}(α + ϵ, z(ϵ)) = (α + ϵ, \frac{a(α+ϵ)+b(α+ϵ)t(ϵ)}{c(α+ϵ)+d(α+ϵ)t(ϵ)}) which, clearly specialises to (α, \frac{a(α)}{c(α)}), using the fact that c(α) ≠ 0. The lemma is then shown.

□

Lemma 3.6. Effect on parabolic branches at Q_{1}

Let C_{1}, C_{2} and θ_{g} satisfy the conditions of the previous lemma and suppose that γ is a parabolic branch of C_{1}, centred at Q_{1}. Then the corresponding branch θ_{g}(γ) is centred at;

[1 : 0 : 0] if max{ord(a(x)), ord(b(x))} << min{ord(c(x)), ord(d(x))}

[0 : 1 : 0] if max{ord(c(x)), ord(d(x))} << min{ord(a(x)), ord(b(x))}

Proof. As γ is parabolic and centred at Q_{1}, given an infinitesimal ϵ ∈ V_{0}, there exists z(ϵ) such that (\frac{1}{ϵ}, z(ϵ)) ∈ (V_{Q_{1}} ∩ C_{1} ∩ γ). By a straightforward calculation, we can assume that z(ϵ) is of the form \frac{u(ϵ)}{ϵ^{i}}, for a unit u(x) ∈ L[[x^{\frac{1}{r}}]], r ≥ 1, and 1 < i ≤ N, i ∈ Q, where N depends only on the degree of C_{1}. We have that;

g(\frac{1}{ϵ}, z(ϵ)) = \frac{a( \frac{1}{ϵ})u(ϵ)+b( \frac{1}{ϵ})ϵ^{i}}{c( \frac{1}{ϵ})u(ϵ)+d( \frac{1}{ϵ})ϵ^{i}} = q(ϵ)

and

θ_{g}(\frac{1}{ϵ}, z(ϵ)) = [\frac{1}{ϵ} : q(ϵ) : 1] = [1 : ϵq(ϵ) : ϵ], (⋆)

In the case that ord(q(ϵ)) > −1, the specialisation, determined from (⋆), is given by [1 : 0 : 0], and, in the case that ord(q(ϵ)) < −1, the specialisation is given by [0 : 1 : 0]. By inspection of q(ϵ), these conditions are clearly met, given the corresponding conditions of the lemma.

□

It is convenient, in what follows, to consider the effect of the following transformation;

Lemma 3.7. Let C_{1} satisfy the conditions of the previous lemma, and let α ≠ 0 be chosen so that x = α does not correspond to the tangent line of any hyperbolic branch of C_{1}, centred at Q_{1}. Then the transformation;

θ_{g} : (x, z) ↦ (x, (x − α)z)

has the following properties;

(i). θ_{g} defines a birational map θ_{g} : C_{1} ↭ C_{2}, with C_{2} satisfying the conditions of Lemma 3.3.

(ii). If γ is a branch of C_{1}, centred along (l_{∞} \ [1 : 0 : 0]), then θ_{g}(γ) is centred at Q_{1} ∈ C_{2}.

Moreover,

(iii). If γ is a branch of C_{1}, centred at [1 : 0 : 0], then, after finitely many repetitions of transformations, having the form θ_{g}, we obtain a birational map θ : C_{1} ↭ C_{2}, such that θ(γ) is centred at Q_{1} ∈ C_{2}.

(iv). In particular, C_{2} intersects the line l_{∞} only at Q_{1}.

Proof. The property (i) follows immediately from Lemma 3.4 and the choice of α ≠ 0. By explicit calculation, the projective formulation of θ_{g} is given by;

θ_{g} : [X : Z : W] ↦ [XW : XZ − αW Z : W^{2}] (⋆⋆)

Any point P, lying on (l_{∞} \ (Q_{1} ∪ [1 : 0 : 0]), satisfies W = 0 and XZ ≠ 0, hence, by (⋆⋆), is mapped to Q_{1}. In particular, if γ is a branch of C_{1}, centred along (l_{∞} \ (Q_{1} ∪ [1 : 0 : 0]), then θ_{g}(γ) is centred at Q_{1}. By the choice of α in relation to tangent lines, and Lemma 3.5, the hyperbolic branches of C_{1}, centred at Q_{1}, are fixed. One finds the effect on the parabolic branches of C_{1}, centred at Q_{1}, by explicit calculation. Using the notation of Lemma 3.6, for z(ϵ) = \frac{u(ϵ)}{ϵ^{i}}, with i > 1, we have that;

θ_{g}(\frac{1}{ϵ}, z(ϵ)) = [ϵ^{i} : u(ϵ) − αu(ϵ)ϵ : ϵ^{i+1}], (⋆ ⋆ ⋆)

Clearly, the specialisation of this point is Q_{1} = [0 : 1 : 0]. Hence, the parabolic branches of C_{1}, centred at Q_{1}, are also fixed. (ii) is then shown.

Again, we find the effect on a branch γ, centred at [1 : 0 : 0], by explicit calculation. Using the notation of Lemma 3.6, for z(ϵ) = \frac{u(ϵ)}{ϵ^{i}}, with i < 1, by considering (⋆ ⋆ ⋆), we have that θ_{g}(γ) is centred along (l_{∞} \ [1 : 0 : 0]) iff 0 ≤ i < 1. Otherwise, we obtain a new z_{1}(ϵ) = \frac{u_{1}(ϵ)}{ϵ^{i_{1}}}, with i_{1} = i + 1. Repeating the calculation, for new transformations of the form θ_{g}, satisfying the previous conditions of the lemma, we are guaranteed that this process eventually produces a birational map θ : C_{1} ↭ C_{2}, with θ(γ) centred along (l_{∞} \ [1 : 0 : 0]). Using the previous properties (i) and (ii), we can assume that θ(γ) is centred at Q_{1}. Hence, (iii) is shown.

The final statement follows straightforwardly from (i), (ii) and (iii).

□

We now consider the following method of moving branches in finite position, not situated along the axis x = 0, to Q_{1};

Lemma 3.8. Let C_{1}, C_{2}, g and θ_{g} satisfy the conditions of Lemma 3.4, with the additional property that, for a given α ≠ 0, c(α) = d(α) = 0, a(α) ≠ 0 and a(α)β + b(α) ≠ 0, for any of the finitely many branches of C_{1} in finite position, centred along the axis x = α, with coordinates (α, β). Then, if γ is a branch of C_{1}, centred in finite position along the axis x = α, θ_{g}(γ) is centred at Q_{1} ∈ C_{2}, and, any hyperbolic branch of C_{1}, centred at Q_{1} with tangent line x = α, is fixed.

Proof. The case when γ is centred in finite position, with coordinates (α, β), such that a(α)β + b(α) ≠ 0 follows by a direct calculation. The statement, concerning hyperbolic branches, follows immediately from Lemma 3.5. □

As a consequence of the previous arguments, we have the following;

Lemma 3.9. There exists an irreducible plane projective curve C′ ⊂ P^{2}, satisfying the conditions of Lemma 3.3, with the additional properties;

(i). (C′ ∩ l_{∞}) = Q_{1}.

(ii). All the singularities of C′ lie along the axis x = 0.

Proof. By Lemma 3.7(ii), we can assume there exists a curve C_{1}, satisfying the conditions of Lemma 3.3, with the additional property that (C_{1} ∩ l_{∞}) = {Q_{1}}. In particular, all the singularities of (C_{1} \ {Q_{1}}) are in finite position. Let A = {α_{1}, …, α_{j}, …, α_{r}} enumerate the axes x = α_{j}, containing the singularities in finite position, away from x = 0. Let B = {α_{r+1}, …, α_{k}, …, α_{s}} enumerate the axes x = α_{k}, corresponding to tangent lines of hyperbolic branches, centred at Q_{1}, away from x = 0, not contained in A. Let C_{j} = {β_{j1}, …, β_{ji}, …, β_{jm}}, for 1 ≤ j ≤ r, 1 ≤ i ≤ m, where deg(C_{1}) = m, enumerate the zcoordinates of the intersections in finite position of C_{1} with the axis x = α_{j}. It is an elementary exercise to show, using the fact that 0 ∉ (A∪B), that there exist polynomials {a(x), b(x), c(x), d(x)} ∈ L[x], with the properties;

(a). a(α_{1}) … a(α_{r}) … a(α_{s}) ≠ 0.

(b). c(α_{1}) = … = c(α_{r}) = … = c(α_{s}) = 0.

(c). d(α_{1}) = … = d(α_{r}) = 0.

(d). b(α_{j}) ≠ −a(α_{j})β_{ji}, 1 ≤ j ≤ r, 1 ≤ i ≤ m.

(e). a(0)d(0) ≠ 0 and b(0) = 0.

(f). max{ord(c(x)), ord(d(x))} << min{ord(a(x)), ord(b(x))}.

By (e) and Lemma 3.4, these polynomials determine a birational morphism θ_{g} : C_{1} ↭ C_{2}, such that C_{2} also satisfies the conditions of Lemma 3.3. By (a),(b),(c),(d) and Lemma 3.8, all the singularities of C_{1}, in finite position, away from the axis x = 0, are moved to Q_{1} ∈ C_{2}. By (a),(b) and Lemma 3.5, the position of all the hyperbolic branches of C_{1} at Q_{1} is fixed. By (f) and Lemma 3.6, the position of all the parabolic branches of C_{1} at Q_{1} is fixed. Denoting C_{2} by C′, it is then clear that conditions (i) and (ii) of the lemma are satisfied for C′.

□

We recall from [5] that the notion of a branch is invariant under birational transformations. The following result allows us to move any finite number of branches away from the critical point Q_{1}.

Lemma 3.10. Let C_{1} satisfy the conditions of Lemma 3.3 and let {γ_{1}, …, γ_{t}} enumerate a finite subset of its branches. Then there exists a birational map θ_{g} : C_{1} ↭ C_{2}, with C_{2} also satisfying the conditions of Lemma 3.3, such that the corresponding branches {θ_{g}(γ_{1}), …, θ_{g}(γ_{t})} are either in finite position or centred at the point [1 : 0 : 0] along the axis l_{∞}. In particular, none of the branches are centred at the point Q_{1}.

Proof. By birationality, we can assume that C_{1} satisfies the conditions of Lemma 3.9. Given the above enumeration, let; (i). {γ_{1}, …, γ_{r}} consist of those in finite position.

(ii). {γ_{r+1}, …, γ_{s}} consist of the hyperbolic branches, centred at Q_{1}.

(iii). {γ_{s+1}, …, γ_{t}} consist of the parabolic branches, centred at Q_{1}.

Let {(α_{1}, β_{1}), …, (α_{j}, β_{j}), …, (α_{r}, β_{r})} enumerate the coordinates of the branches given in (i). Let {δ_{r+1}, …, δ_{k}, …, δ_{s}} enumerate the axes x = δ_{k}, appearing as tangent lines to the branches given in (ii). It is an elementary exercise to show that there exist polynomials {a(x), b(x), c(x), d(x)} ∈ L[x], with the properties;

(a). c(α_{1}) … c(α_{j}) … c(α_{r})c(δ_{r+1}) … c(δ_{k}) … c(δ_{s}) ≠ 0.

(b). a(α_{j})β_{j} + b(α_{j}) = β_{j}, c(α_{j})β_{j} + d(α_{j}) = 1, (1 ≤ j ≤ r)

(c). a(0)d(0) ≠ 0 and b(0) = 0.

(d).max{ord(a(x)), ord(b(x))} << min{ord(c(x)), ord(d(x))}.

By (c) and Lemma 3.4, there exists a corresponding birational map θ_{g} : C_{1} ↭ C_{2}, with C_{2} also satisfying the conditions of Lemma 3.3. By (b), the branches {θ_{g}(γ_{1}), …, θ_{g}(γ_{r})} are all centred at the same points as {γ_{1}, …, γ_{r}} respectively. In particular, the branches from (i) remain in finite position. By (a) and Lemma 3.5, the branches {θ_{g}(γ_{r+1}), …, θ_{g}(γ_{s})} are centred at the points in finite position;

\{(δ_{r+1}, \frac{a(δ_{r+1})}{c(δ_{r+1})}), …, (δ_{s}, \frac{a(δ_{s})}{c(δ_{s})})\}

respectively. In particular, the branches from (ii) are moved to finite position. By (d) and Lemma 3.6, the branches {θ_{g}(γ_{s+1}), …, θ_{g}(γ_{t})} are centred at the point [1 : 0 : 0]. Hence, the lemma is shown.

□

We now make the following definition;

Definition 3.11. Fixing an enumeration of branches for some curve C′, satisfying the conditions of Lemma 3.3, we define a branch γ to be blue, if there exists a birational map θ : C ↭ C_{1}, with the curve C_{1} also satisfying the conditions of Lemma 3.3, such that; Either

(i). θ(γ) is centred in finite position (a, b), with I_{γ}(C_{1}, x = a) ≥ 2.

Or

(ii). θ(γ) is centred at [1 : 0 : 0], with I_{γ}(C_{1}, l_{∞}) ≥ 2.

where I_{γ} is defined as in [6]. We define a branch γ to be silver otherwise.

We observe the following;

Remarks 3.12. (a). The choice of colours is partly motivated by consideration of the optical spectrum of visible light, as well as theoretical views of the author on chemical processes that occur in the production of light by stars. I hope to explain my views on the connections between geometry and a 4-fold model of the theory of light in a forthcoming paper.

(b). On a geometrical level, the above construction is well-defined. For suppose that C_{1} and C_{2} are both curves, for which Definition 3.11 is made, relative to C′. Then there exists a birational map θ_{1} : C_{1} ↭ C_{2}, for which (pr_{1} ∘ θ_{1}) = pr_{1}, (⋆). It is a simple exercise to check, by (⋆), the definition of I_{γ} using infinitesimals, see [6], and birationality of the map θ_{1}, that, if γ is a branch of C_{1}, centred at (a, b), with θ(γ) centred at (a, b′), then I_{γ}(C_{1}, x = a) = I_{θ(γ)}(C_{2}, x = a). Similarly, if γ is a branch of C_{1}, centred at [1 : 0 : 0], with θ(γ) centred at [1 : 0 : 0], then I_{γ}(C_{1}, l_{∞}) = I_{θ(γ)}(C_{2}, l_{∞}).

(c). Using Lemmas 3.5, 3.6 and 3.9, any branch γ of C′ may be centred in finite position or at [1 : 0 : 0], by an appropriate birational map θ. Hence, we can give an equivalent characterisation of a silver branch as in Definition 3.11, replacing the either/or clause by;

Either

(i). θ(γ) is centred in finite position (a, b), with I_{γ}(C_{1}, x = a) = 1.

Or (ii). θ(γ) is centred at [1 : 0 : 0], with I_{γ}(C_{1}, l_{∞}) = 1.

On an intuitive level, this means that a silver branch is characterised by the condition that it is non-singular and transverse to a vertical axis or to the line l_{∞}, when centred away from Q_{1}. Similarly, a blue branch is characterised by the condition that it is either singular or tangent to a vertical axis/the line l_{∞}, when centred away from Q_{1}.

(d). There exist infinitely many silver branches and finitely many blue branches belonging to C′. This follows immediately from the above characterisation and the fact that, there exists a unique non-singular branch, transverse to the axis x = a, for a generic point (a, b) belonging to C′. Clearly, there exist finitely many silver branches and finitely many blue branches, belonging to C′, centred along the axis x = 0.

We now show the following;

Lemma 3.13. Uniformity of branches

Let C′ satisfy the conditions of Lemma 3.3. Then the distribution of silver and blue branches along any axis is uniform. More precisely;

(i). If x = α is a vertical axis, then, either, every branch centred in finite position along x = α is silver, or, every branch centred in finite position along x = α is blue.

(ii). Either, every branch centred along l_{∞} \ {Q_{1}} is silver, or, every branch centred along l_{∞} \ {Q_{1}} is blue.

(iii). If x = α is a vertical axis, then, either, every hyperbolic branch, centred at Q_{1}, tangent to the line x = α, is silver, or, every hyperbolic branch, centred at Q_{1}, tangent to the line x = α, is blue.

(iv). Either, every parabolic branch, centred at Q_{1}, is silver, or, every parabolic branch, centred at Q_{1}, is blue.

Proof. It is elementary that, for generic δ, the curve C′ intersects the axis x = δ transversely at t distinct points in finite position, where t = deg(L(C′)/L(x)). As, by Lemma 3.3(iv), the extension L(C′)/L(x) is Galois, we have that |G| = t, where G = Gal(L(C′)/L(x)), and G acts transitively on the fibre (C′ ∩ (x = α)) \ {Q_{1}}, (⋆). As this condition is definable, we can find an open subset U ⊂ A^{1}, with the property that, for any δ ∈ U, (⋆) holds for the fibre x = δ. Suppose that there exists a silver branch γ_{1} and a blue branch γ_{2} situated on the axis x = α, centred at the coordinates (α, β_{1}) and (α, β_{2}) respectively. Choose an infinitesimal ϵ ∈ V_{0}, such that α + ϵ ∈ U, and (α + ϵ, β′_{1}) ∈ γ_{1}, (α + ϵ, β′_{2}) ∈ γ_{2}. As (⋆) holds for the axis x = α + ϵ, we can find a birational morphism θ_{h} : C′ ↭ C′, representing the action of h ∈ G, with pr∘θ_{h} = pr, and θ_{h}(α+ϵ, β′_{1}) = (α+ϵ, β′_{2}). It follows, by construction, that θ_{h}(γ_{1}) = γ_{2}. However, by previous remarks 3.12(b), we have that I_{θ_{h}(γ_{1})}(C′, x = α) = I_{γ_{1}}(C′, x = α) = 1, contradicting the fact that θ_{h}(γ_{1}) = γ_{2} is a blue branch. Hence, (i) is shown. In order to show (ii), one should employ a similar argument, using the fact that, for ϵ ∈ V_{0}, with ϵ ≠ 0, x = \frac{1}{ϵ} lies in U. For (iii) and (iv), one can use Lemmas 3.5, Lemmas 3.6 and Definition 3.11, to reduce to the cases (i) and (ii) respectively.

□

Lemma 3.14. Separation of silver branches

Let C′ satisfy the conditions of Lemma 3.3, then any silver branch γ of C′, not centred along the axis x = 0, (^{5}), can be isolated away from Q_{1}. That is, there exists a birational map θ : C′ ↭ C_{1}, with C_{1} also satisfying the conditions of Lemma 3.3, such that;

Either (i). θ(γ) is centred at a point (α, β) in finite position, with α ≠ 0, and is the unique branch of C_{1} to be centred at this point.

Or (ii). θ(γ) is centred at a point of (l_{∞} \ {Q_{1}}), and is the unique branch of C_{1} to be centred at this point.

Moreover, given any finite set {γ_{1}, …, γ_{r}} of silver branches, centred at Q_{1}, with distinct tangent directions, not including x = 0, there exists a birational map θ : C′ ↭ C_{1}, with C_{1} also satifying the conditions of Lemma 3.3, such that {θ(γ_{1}), …, θ(γ_{r})} are all isolated away from Q_{1}.

Proof. By Lemmas 3.5 and 3.6, we can assume that the silver branch γ is centred either in finite position or at [1 : 0 : 0].

Case 1. Assume that γ is centred at (α, β), with α ≠ 0.

5We follow the convention that a branch, centred along x = 0, includes hyperbolic branches, centred at Q_{1}, with tangent line x = 0.

By Lemma 3.13, all the branches situated along the axis x = α are silver. In particular, all the branches centred at (α, β) are silver. Let {γ, γ_{1}, …, γ_{i}, …, γ_{r}} be an enumeration of these branches. By results of [6], see also Remarks 2.6, we can find parametrisations of the form;

{(α + x, β + η(x)), (α + x, β + η_{1}(x)), …, (α + x, β + η_{r}(x))}

for this enumeration, where {η(x), η_{1}(x), …, η_{i}(x), …, η_{r}(x)} are distinct power series, belonging to L[[x]], of the form;

η(x) = ∑_{j=1}^{∞} c_{j}x^{j}, η_{i}(x) = ∑_{j=1}^{∞} c_{ij}x^{j}, 1 ≤ i ≤ r

Let g(x, z) be defined by;

g(x, z) = \frac{(z−β)}{(x−α)} − \frac{β}{α} = \frac{(αz−βx)}{α(x−α)}, (⋆)

We have that α.α(x − α)|_{(0,0)} = −α^{3} ≠ 0 and (−βx)|_{(0,0)} = 0, hence, by Lemma 3.4, g(x, z) defines a birational map θ_{g} : C′ ↭ C′′, with C′′ also satisfying the conditions of Lemma 3.3. Using Lemma 3.8, if γ′ is a branch of C′ in finite position, along the axis x = α, not at (α, β), then θ_{g}(γ) is centred at the point Q_{1} ∈ C′′. By Lemma 3.5, if γ′ is a hyperbolic branch of C′, centred at Q_{1}, with tangent line x = α, then θ(γ′) is also centred at Q_{1} ∈ C′′. It follows that the only branches, in finite position along the axis x = α of C′′, are {θ_{g}(γ), θ_{g}(γ_{1}), …, θ_{g}(γ_{r})}. We now consider the positions of these branches;

By results of [6] and the definition of g(x, z), the branches {θ_{g}(γ), θ_{g}(γ_{1}), …, θ_{g}(γ_{r})} are parametrised by;

\{(α + x, δ(x) − \frac{β}{α}), (α + x, δ_{1}(x) − \frac{β}{α}), …, (α + x, δ_{r}(x) − \frac{β}{α})\}

where η(x) = α + xδ(x) and η_{i}(x) = α + xδ_{i}(x), for 1 ≤ i ≤ r. It follows that θ_{g}(γ) is centred in finite position, at (α, β_{1}), where β_{1} depends only on the first coefficient c_{1} of the power series η(x). In particularly, a branch θ_{g}(γ_{i}), for 1 ≤ i ≤ r, is also centred at (α, β_{1}), if and only if, c_{i1} = c_{1}. We then repeat the above construction for the new centre (α, β_{1}), obtaining a new birational map θ_{g_{1}}. The only branches remaining in finite position along the axis x = α after the composition (θ_{g_{1}} ∘ θ_{g}), being those for which c_{i1} = c_{1}, and, the only branches whose centre coincides with that of (θ_{g_{1}} ∘ θ_{g})(γ), being those for which c_{i1} = c_{1} and c_{i2} = c_{2}. As the power series η(x) is distinct from {η_{1}(x), …, η_{r}(x)}, clearly the construction terminates after a finite number of steps, and, we obtain a curve C_{1}, satisfying the conditions of Lemma 3.3, together with a birational map θ : C ↭ C_{1}, such that θ(γ) is the only branch, centred in finite position, along the axis x = α. Hence, (i) is shown for this case.

Case 2. Assume that γ is centred at [1 : 0 : 0].

Choosing α ≠ 0, we consider the rotation ν about Q_{1}, defined by;

ν : (x, z) ↦ (\frac{1}{x+α} − \frac{1}{α}, z) = (\frac{−x}{α(x+α)}, z)

By construction, we have that ν(0, 0) = (0, 0) and ν fixes the axis x = 0. It is clear that ν determines a birational morphism θ_{ν} : C′ ↭ C′′, which is etale in an open neighborhood of (0, 0), (^{6}). It is a straightforward exercise, using infinitesimals, and the fact that ν permutes the vertical axes, to check that θ_{ν} interchanges;

(a). The silver branches of C′, centred along (l_{∞} \ ([1 : 0 : 0] ∪ Q_{1})), and the silver parabolic branches of C′, centred at Q_{1}, with the silver hyperbolic branches of C′′, centred at Q_{1}, with tangent line x = −\frac{1}{α}.

(b). The silver branches of C′, centred at [1 : 0 : 0], with the silver branches of C′′, centred in finite position along the axis x = −\frac{1}{α}. (⋆⋆)

We now apply Case 1, to obtain a birational morphism θ_{1} : C′′ ↭ C′′′, in the stronger sense of Lemma 3.4, such that the silver branch θ_{ν}(γ) is isolated at (−\frac{1}{α}, 0), that is (θ_{1} ∘ θ_{ν})(γ) is the unique branch of C′′′ to be centred at (−\frac{1}{α}, 0), (⋆ ⋆ ⋆).

Then, we apply the inverse rotation ν^{−1} about Q_{1}, defined by;

ν^{−1} : (x, z) ↦ (\frac{1}{x+ \frac{1}{α}} − α, z) = (\frac{−α^{2}x}{αx+1}, z)

which determines a birational map θ_{ν^{−1}} : C′′′ ↭ C′′′′. It is simple to check that the composition (θ_{ν^{−1}} ∘ θ_{1} ∘ θ_{ν}) : C′ ↭ C′′′′ is a birational map, in the stronger sense of Lemma 3.4. Using (⋆⋆), we have that (θ_{ν^{−1}} ∘ θ_{1} ∘ θ_{ν})(γ) is centred at [1 : 0 : 0]. If (\frac{−1}{α} + x, δ(x)), with δ(x) ∈ L[[x]] and ord(δ(x)) ≥ 1, is a parametrisation of the silver branch θ_{1} ∘ θ_{ν})(γ), then the corresponding parametrisation of (θ_{ν^{−1}} ∘ θ_{1} ∘ θ_{ν})(γ) is given by (\frac{1}{x} − α, δ(x)). Moreover, using (⋆⋆)(b) and (⋆ ⋆ ⋆), if (\frac{1}{x} − α, δ_{1}(x)) is a parametrisation of any other silver branch of C′′′′, centred at [1 : 0 : 0], then we must have that ord(δ_{1}(x)) = 0, (⋆⋆⋆⋆). We, finally, apply a transformation of the form θ_{g} : C′′′′ ↭ C′′′′′, considered in Lemma 3.7. By (⋆⋆⋆⋆), it is elementary to check, using infinitesimals, that the branch (θ_{g} ∘ θ_{ν^{−1}} ∘ θ_{1} ∘ θ_{ν})(γ) is fixed at [1 : 0 : 0] and the other branches of C′′′′, centred at [1 : 0 : 0], are moved to positions along the axis (l_{∞} \ (Q_{1} ∪ [1 : 0 : 0])), see also the proof of Lemma 3.7. By the properties of θ_{g}, considered in Lemma 3.7, (θ_{g} ∘ θ_{ν^{−1}} ∘ θ_{1} ∘ θ_{ν})(γ) is the unique branch of C′′′′′ to be centred at [1 : 0 : 0]. Letting θ = (θ_{g} ∘ θ_{ν^{−1}} ∘ θ_{1} ∘ θ_{ν}) and C_{1} = C′′′′′, we have that (ii) is shown for this case.

6This is not a morphism in the sense of Lemma 3.4 and footnote 3, as, clearly, (pr ∘ θ) ≠ pr

For the final part of the lemma, assume, first, that all of the branches {γ_{1}, …, γ_{r}} are hyperbolic. By Lemma 3.10, we can assume that they are all in finite position. Let {α_{1}, …, α_{r}} enumerate the distinct xcoordinates of these branches. Using Case 1 above, we can isolate the branch γ_{1}, away from Q_{1}, by applying finitely many transformations of the form, given in (⋆), with α = α_{1}. Clearly, the branches {γ_{2}, …, γ_{r}} remain in finite position after these transformations. Suppose, inductively, that we have isolated the branches {γ_{1}, …, γ_{i}}, with 1 ≤ i < r, away from Q_{1}. We can then apply finitely many transformations of the form, given in (⋆), with α = α_{i+1}, in order to isolate the branch γ_{i+1}, away from Q_{1}. As the restriction of these transformations to the axes {x = α_{1}, …, x = α_{i}} is of the form a(α_{j})z + b(α_{j}), with a(α_{j}) = (α_{i+1} − α_{j})^{−1} ≠ 0, for 1 ≤ j ≤ i, the branches {γ_{1}, …, γ_{i}} also remain isolated away from Q_{1}. Hence, we have isolated the branches {γ_{1}, …, γ_{i}, γ_{i+1}}. By induction, we can, therefore, isolate all the branches {γ_{1}, …, γ_{r}}, away from Q_{1}, as required, (⋆ ⋆ ⋆ ⋆ ⋆). Assume, now, that {γ_{1}, …, γ_{r}} are hyperbolic branches and γ_{r+1} is parabolic. By Lemma 3.10, we can assume that {γ_{1}, …, γ_{r}} are all centred in finite position, and γ_{r+1} is centred at [1 : 0 : 0]. By (⋆ ⋆ ⋆ ⋆ ⋆), we can isolate the branches {γ_{1}, …, γ_{r}}, away from Q_{1}, using finitely many transformations of the type considered in (⋆). By a simple projective calculation, such transformations fix the point [1 : 0 : 0], in particular the branch γ_{r+1} remains centred at [1 : 0 : 0]. We now follow the method of Case 2, in order to isolate the branch γ_{r+1}, away from Q_{1}, while, retaining the property (⋆ ⋆ ⋆ ⋆ ⋆⋆), that the remaining branches {γ_{1}, …, γ_{r}} are isolated, away from Q_{1}, in finite position. This amounts to showing that (⋆⋆⋆⋆⋆⋆) is preserved in the following steps from Case 2;

(1). For the rotation ν, with α ≠ α_{i}, for 1 ≤ i ≤ r.

(2). For the construction of the birational map θ_{1}, with the additional use of the previous inductive proof.

(3). For the inverse rotation ν^{−1}, with α ≠ α_{i}, for 1 ≤ i ≤ r.

(4). For transformations of the form θ_{g}, considered in Lemma 3.7, with α ≠ α_{i}, for 1 ≤ i ≤ r.

Hence, the final part of the lemma is shown.

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Remarks 3.15. This seems to be a reasonable geometrical property of a theory of light, motivated by theoretical considerations of the author on the chemical production of energy by fission in stars, see also Remarks 3.12(a). The increased level of radioactivity along the axis x = 0 is, in fact, an observable phenomenon of pulsars, a type of neutron star, see [11].

The following result is straightforward, see the previous footnote 3;

Lemma 3.16. Let C_{1} and C_{2} satisfy the conditions of Lemma 3.3, then there exists a birational map θ : C_{1} ↭ C_{2}, which is an isomorphism in the etale topology with (0, 0) as labelled points, and (pr ∘ θ) = θ.

Proof. By the definition of the etale topology, there exists an irreducible projective algebraic curve (C_{3}, ∗), together with an open subset U_{3} ⊂ C_{3}, containing the marked point ∗, and open subsets U_{1} ⊂ C_{1}, U_{2} ⊂ C_{2}, containing (0, 0), with etale morphisms;

i_{13} : (C_{3}, U_{3}, ∗) → (C_{1}, U_{1}, (0, 0))

i_{23} : (C_{3}, U_{3}, ∗) → (C_{2}, U_{2}, (0, 0))

such that i_{13}^{∗}(δ_{i}(x)) = i_{23}^{∗}(δ_{i}(x)), for 1 ≤ i ≤ m, (∗), and (pr ∘ i_{13}) = (pr∘i_{23}), (∗∗). Using (∗∗), the morphisms i_{13} and i_{23} induce inclusions;

i_{13}^{∗} : L(C_{1}) → L(C_{3}) i_{23}^{∗} : L(C_{2}) → L(C_{3})

with (i_{13}^{∗} ∘ pr^{∗}) = (i_{23}^{∗} ∘ pr^{∗}) : L(x) → L(C_{3}), (∗ ∗ ∗). By (∗), (∗ ∗ ∗), and Lemma 3.3(iii), we can define an isomorphism ϕ^{∗} : L(C_{2}) → L(C_{1}), such that (ϕ^{∗} ∘ pr^{∗}) = pr^{∗} : L(x) → L(C_{1}). Hence, there exists a birational map ϕ : C_{1} ↭ C_{2}, with (pr ∘ ϕ) = pr, (∗ ∗ ∗∗). By Lemma 3.3(i), there exists a unique silver branch γ_{1} of C_{1}, centred at the origin (0, 0). Using (∗ ∗ ∗∗), it is straightforward that θ(γ_{1}) is a silver branch of C_{2}, centred along the axis x = 0. Using Lemma 3.3(iv), and the method of Lemma 3.13, we can find a birational map θ_{h} : C_{2} ↭ C_{2}, with (pr ∘ θ_{h}) = pr, (∗ ∗ ∗ ∗ ∗) representing h ∈ Gal(L(C_{2})/L(x)), such that (θ_{h}∘ϕ)(γ_{1}) = γ_{2}, where γ_{2} is the unique silver branch of C_{2}, centred at (0, 0). It is a simple calculation, using infinitesimals, Theorems (6.10,6.11) of [4] and the fact that the branches {γ_{1}, γ_{2}} are nonsingular, to show that (θ_{h} ∘ϕ) is etale in an open neighborhood of C_{1}, containing (0, 0). Similar considerations apply to the morphism (θ_{h} ∘ ϕ)^{−1}, hence, (θ_{h} ∘ θ) induces an isomorphism in the etale topology, with (0, 0) as labelled points. Letting θ = (θ_{h} ∘ ϕ), the fact that (pr ∘ θ) = pr follows immediately from (∗ ∗ ∗∗) and (∗ ∗ ∗ ∗ ∗). Hence, the lemma is shown.

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Remarks 3.17. It is a straightforward exercise, to show that a morphism satisfying the conditions of Lemma 3.17 is unique. Hence, we can unambigiously identify the branches of any two curves, satisfying the conditions of Lemma 3.3. The previous Lemmas 3.5 to 3.15, support the intuition that the orbits of each branch resemble silver or blue flashes of light, emanating from and towards a central point Q_{1}. Given the high gravitational field of a neutron star, this is, possibly, a good description of the behaviour of light in certain situations. Remarks 3.18. Rose windows are another geometrically interesting source of examples, that exhibit both radial and rotational symmetry, as well as intricate light effects. Some of the best can be found in the Rayonnant designs of French medieval cathedrals. An excellent account of the historical development of rose windows is given in [2]. For the reader, interested in making rose windows, one should look at [16]. A number of pictures of rose windows should be available soon on the website, http://www.magneticstrix.net, where you can find some good examples of neutron star style windows in the Rayonnant Rose of Saint Denis, and the North and South Roses of Notres Dame cathedral.

We now make the following refinement of the branch terminology that we have used;

36 TRISTRAM DE PIRO

Definition 3.19. Let conditions be as in Definition 3.11, then we define a branch γ to be violet, if;

γ is a non-singular blue branch. and we define a branch γ to be green, if;

γ is a silver branch, centred along the axis x = 0, (see also footnote 5), or γ is a blue branch, but not a violet branch. Remarks 3.20. In order to see that this is a good definition,

We then have the following geometric interpretation of Lemma 2.5. Lemma 3.21. Let hypotheses be as in Lemma 2.5 and notation as in Definition 3.1, then we can find an irreducible closed algebraic surface S ⊂ P^{3}, such that the restriction of pr_{1} to (S \ Q_{1}) defines a quasi-finite morphism, pr_{1} : S → P_{1}^{2}, (♡), with the following further additional properties;

(i). There exist open subsets U ⊂ A_{1}^{2} ⊂ P_{1}^{2} and V ⊂ S∩A^{3} such that; pr_{1} : (V, (00)^{lift}) → (U, (00)) is etale. (ii). There exist distinct irreducible closed algebraic curves {C_{1}, …, C_{m}}

contained in S, such that; a. pr_{1} : (C_{j} \ {Q_{1}}) → C is a finite cover, for 1 ≤ j ≤ m, (^{7}) b. \overline{(pr_{1}^{−1}(C) ∩ S)} = ∪_{1≤j≤m}C_{j}. c. The coordinate ring R(V) embeds in L[[x]][y] ∩ L(x, y)^{alg} and the

defining equations of C_{j} ∩ V are given by; y − η_{j}(x) = 0, for 1 ≤ j ≤ m. (iii). There exists an irreducible algebraic curve, C′ ⊂ P_{2}^{2}, satisfy-

ing the conditions of Lemma 3.3, with respect to the coordinate system

7We use the term cover, in the sense that pr_{1} defines a dominant morphism from (C_{j} \ {Q_{1}}) onto an open subset of C. The projection naturally extends to define a surjective map from the branches of C_{j}, including those centred at Q_{1}, to the branches of C. The reader should look at the paper [6]

FLASH GEOMETRY OF ALGEBRAIC CURVES 37

(x, z), such that; a. pr_{2} : C_{j} → C′ is a finite cover, for 1 ≤ j ≤ m. b. pr_{2} : C_{j} ↭ C′ is a birational map, with inverse η_{j} : C′ ↭ C_{j} and an open subset V′ ⊂ C′, such that; c. pr_{1} : (V′, (00)) → (A^{1}, 0) is a Galois cover, with the extension

L(V′)/L(x) equal to the Galois closure of the extension L(C)/L(x). Proof. From the previous construction of Lemma 2.5, the power series {δ_{1}(x), …, δ_{m}(x)} belong to L[[x]] ∩ L(x)^{alg}. By Theorem 1.3 and remarks at the beginning of Section 3 of [5], we have that L[[x]] ∩ L(x)^{alg} is isomorphic to the local ring for the etale topology O_{(A^{1},0)}^{et}. By the construction in Lemma 3.3, we can find an irreducible plane projective curve C′ ⊂ P^{2}, such that;

(i). There exists an open subset U_{1} ⊂ C′, such that (0, 0) ∈ U_{1} and; pr : (U_{1}, (0, 0)) → (A^{1}, 0) is an etale morphism, (†) (ii). {δ_{1}(x), …, δ_{m}(x)} all belong to R(U_{1}), for a given embedding

of R(U_{1}) in L[[x]] ∩ L(x)^{alg}, and δ_{1}(0, 0) = … = δ_{m}(0, 0) = 0, (†′)

(iii). The function field L(U_{1}) is generated over L(x) by the power series {δ_{1}(x), …, δ_{m}(x)}, (^{8}). (†′′).

By Fact 1.5 of [5], we can find an open subset U′ ⊂ A^{1}, containing (0), and a monic polynomial F (z) ∈ R(U′)[z] such that the etale morphism pr can be presented, for an open neighborhood V′ ⊂ U_{1} ⊂ C′ of (0)^{lift} = (0, 0) in the form;

Spec((\frac{R(U′)[z]}{F (z)})_{d}) → Spec(R(U′)) (∗)

with F′(z) invertible in (\frac{R(U′)[z]}{F (z)})_{d}. As (V′, (0, 0)) is a localisation of (U_{1}, (0, 0)), the coordinate ring R(U_{1}) ⊂ R(V′) ⊂ L[[x]] ∩ L(x)^{alg} and the conditions (†), (†′) and (†′′) are preserved, replacing U_{1} by V′.

8In non-zero characteristic, one should take L(U_{1}) to be the seperable closure of L(x, δ_{1}(x), …, δ_{m}(x))

Let G(x, z) define the projective curve C′, restricted to A_{2}^{2}, see Definition 3.1, which we also denote by C′. The restriction of C′ to V′ = (pr_{1}^{−1}(U′) ∩ (G = 0) ∩ (d ≠ 0)) corresponds to the cover defined in (∗). We also consider G(x, z) as defining an irreducible algebraic surface C′ × A^{1} in A^{3}, using the coordinates (x, y, z). We now projectivize the affine polynomial G(x, z), by making the substitutions \{x = \frac{X}{W} , z = \frac{Z}{W} \}, and obtain an irreducible homogeneous polynomial R(X, Z, W). This defines an irreducible algebraic surface S ⊂ P^{3}. Observe that, as C′ passes through (0, 0), the surface S contains the point Q_{2}. It is clear from the construction that pr_{2}(S \ {Q_{2}}) = C′ and pr_{1}(S \ Q_{1}) ⊂ P_{1}^{2}. It is straightforward to see that the fibres of pr_{1} restricted to (S \ Q_{1}) are all finite, showing (♡) in the statement of the lemma.

We now set U = U′ × A^{1} ⊂ A_{1}^{2} and V = V′ × A^{1} ⊂ S ∩ A^{3}. By the fact that R(V′) embeds in L[[x]] ∩ L(x)^{alg}, we have that R(V) embeds in L[[x]][y] ∩ L(x, y)^{alg}, (∗∗). The etale morphism pr_{1} : (V′, (00)) → (U′, (0)) may be presented in the form;

R(U′) → \frac{R(U′)[z,w]}{<F (z),wδ(z)−1>} with \frac{∂(F (z),wδ(z)−1)}{∂(z,w)}|_{p} ≠ 0, for p ∈ V′

The morphism pr_{1} : (V, (000) → (U, (00)) may then be presented in the same form, replacing the ring R(U′) by the ring R(U) = R(U′)[y]. By the definition of an etale morphism in Definition 1.1 of [5], we obtain immediately that this morphism is etale, hence (i) of the Lemma is shown.

By (†′) and (∗∗), we now have that the algebraic power series {y−δ_{1}(x), …, y−δ_{m}(x)} define irreducible algebraic curves {D_{1}, …, D_{m}} on V, passing through (00)^{lift} = (000). By the construction of V, we have that the algebraic power series {y−a_{1} −δ_{1}(x), …, y−a_{m} −δ_{m}(x)} define distinct irreducible algebraic curves {D′_{1}, …, D′_{j}, …, D′_{m}} on V ⊂ S, passing through (0, a_{j}, 0), for 1 ≤ j ≤ m. We let C_{j} = \overline{D′_{j}}, for 1 ≤ j ≤ m. By elementary facts on Zariski closure, each C_{j} ⊂ S is irreducible and (ii)(c) of the lemma holds, by the definition of η_{j}(x) = a_{j} + δ_{j}(x) in Lemma 2.5, and the fact that D′_{j} = C_{j} ∩ V. Again, by Lemma 2.5, (pr_{1}^{−1}(C)∩V) = ∪_{1≤j≤m}(C_{j} ∩V), (∗∗∗). It follows immediately, from elementary facts about quasi-finite morphisms, that (ii)(a) of the lemma holds and ∪_{1≤j≤m}C_{j} ⊆ \overline{(pr_{1}^{−1}(C) ∩ S)}. Now suppose that C_{0} ⊂ S is an irreducible component of \overline{(pr_{1}^{−1}(C) ∩ S)}. By elementary dimension theory, C_{0} is an algebraic curve and must define a finite cover of C. We may suppose that (C_{0} ∩A^{3}) ≠ ∅, otherwise C_{0} is contained in the plane (W = 0) and, therefore, pr_{1}(C_{0}) = C is contained in the line l_{1,∞}, contradicting the presentation of C in Lemma 1.1. Suppose that (C_{0} ∩ V) = ∅, then C_{0} ∩ A^{3} must be a line of the form {p} × A^{1}, where p = (p_{1}, p_{2}) is one of the finitely many exceptional points on the affine curve C′ ⊂ A_{2}^{2}, obtained by removing V′. It follows that pr_{1}(C_{0}) = C is contained in the closure of the line x = p_{1}, in the coordinate system (x, y) of A_{1}^{2}, again contradicting the presentation of C in Lemma 1.1. Hence, we may assume that (C_{0} ∩ V) ≠ ∅. It follows that (C_{0} ∩ V) defines an irreducible component curve of (pr^{−1}(C) ∩ V), hence, by (∗ ∗ ∗), coincides with (C_{j} ∩ V), for some 1 ≤ j ≤ m. It follows, by elementary facts on Zariski closure, that C_{0} = C_{j}. Therefore, (ii)(b) of the lemma holds and (ii) of the lemma is shown.

By the presentation given in Lemma 1.1, the point Q_{2} does not lie on C. By (ii)(a) of the lemma, and the fact that Q_{2} is fixed by the projection pr_{1}, Q_{2} cannot lie on C_{j} for any 1 ≤ j ≤ m either. It follows that pr_{2} is defined on all of C_{j}, for 1 ≤ j ≤ m. As we observed earlier, pr_{2}(S \ {Q_{2}}) = C′, hence, as each curve C_{j} belongs to (S \ {Q_{2}}), and, using the argument above to exclude the exceptional case that pr_{2}(C_{j}) is a point, we obtain (iii)(a) of the lemma. Now, observe that the algebraic power series {η_{1}(x), …, η_{m}(x)} define rational functions in L(C′), and belong to the coordinate ring R(V′), for the open subset V′ ⊂ C′. We may, therefore, define a morphism, θ_{j} : V′ → V

θ_{j}(x, z) = (x, η_{j}(x, z), z), for (x, z) ∈ V′

The defining equation of the image of this morphism in R(V) is given by y − η_{j}(x, z) = 0, hence, by (ii)(c) of the lemma, corresponds to (C_{j} ∩ V). It follows that θ_{j} : V′ → C_{j} must define a morphism as well. By the explicit definition of pr_{2} in the affine coordinates (x, y, z), we have that pr_{2} ∘ θ_{j} = Id_{V′}. It follows immediately that pr_{2} defines an isomorphism between the open subsets (C_{j} ∩ V) ⊂ C_{j} and V′ ⊂ C′. Hence, (iii)(b) of the lemma is shown. The property (iii)(c) follows immediately from the construction of V′, by observing that L(V′) = L(x, δ_{1}(x), …, δ_{m}(x)) = L(x, η_{1}(x), …, η_{m}(x)) is a splitting field for the polynomial F (x, y) over L(x), hence, the cover defined in (iii)(c) is Galois and the extension L(V′)/L(x) is equal to the Galois closure of L(C)/L(x), see also footnote 1. Therefore, (iii) of the lemma, is shown. This completes the proof.

□

Definition 3.22. We define the set of curves {C_{1}, …, C_{j}, …, C_{m}}, found in the previous Lemma 3.21, as flashes of the original algebraic curve C, (^{9})

We then have;

Lemma 3.23. There exists a finite group G and an algebraically definable action of G on an open subvariety W′ ⊂ V′, with pr_{1} : W′ → A^{1} a quasi-finite morphism, such that;

(i). For generic x ∈ A^{1}, G acts sharply transitively on the fibre pr_{1}^{−1}(x).

(ii). G induces an algebraically definable, transitive action on the set of flashes {C_{1}, …, C_{m}}.

Proof. We let G = Gal(L(C′)/L(x)). By standard considerations, a given g ∈ G induces a birational morphism θ_{g} : C′ ↭ C′, with the property that pr_{1} ∘ θ_{g} = pr_{1} : C′ ⇝ P^{1}, as rational maps. More specifically, we can find polynomials {p_{g}(x, z), q_{g}(x, z)} with the property that θ_{g} is defined, in affine coordinates, by;

θ_{g} : (x, z) ↦ (x, \frac{p_{g}(x,z)}{q_{g}(x,z)}) = (x, g ▪ z) (∗)

Letting G(x, z) define the curve C′, restricted to the coordinate system (x, z), set R_{g} = ((G = 0) ∩ (q_{g} = 0)), and S_{g} = pr_{1}(R_{g}). Let C′′ define the curve obtained by removing the finitely many coordinate lines {x = s : s ∈ S_{g}}, from the curve C′, realised by G(x, z). It is straightforward to check that C′′ is invariant under the action of θ_{g}. By repeating the construction, we can choose C′′ to be invariant under the action of θ_{g}, for any g ∈ G, (∗∗). If g ∈ G, then C′′ is invariant under the action of both θ_{g} and θ_{g^{−1}} = (θ_{g})^{−1}, it follows that θ_{g} maps C′′ isomorphically onto itself, (∗ ∗ ∗). By a similar argument, removing also coordinate lines of the form {x = t : t ∈ T_{g}}, from V′, where;

9More accurately, a flash C_{j} should be considered as representing the class of curves, corresponding to each power series η_{j}(x). That is, given two curves {C^{1}, C^{2}}, satisfying the conditions of Lemma 3.3, and a birational map α_{12} : C^{1} ↭ C^{2}, as in Lemma 3.16, if {C_{1}^{1}, …, C_{j}^{1}, …, C_{m}^{1}} and {C_{1}^{2}, …, C_{j}^{2}, …, C_{m}^{2}} denote the sets of curves obtained in Lemma 3.21, then we obtain naturally defined birational maps θ_{j} = (η_{j} ∘ α_{12} ∘ (pr_{2})^{−1}) : C_{j}^{1} ↭ C_{j}^{2}. We can then define a flash as the equivalence class of a given curve C_{j} under these birational maps; by Lemma 3.16, this accounts for all such curves.

T_{g} = {t ∈ A^{1} : Card(V′(t)) < Card(V′(x)), for generic x ∈ A^{1}}

we can find an open subvariety W′ ⊂ V′, with the same properties (∗∗) and (∗ ∗ ∗). We consider the action of G on a generic fibre W′(x). Assuming deg_{z}G(x, z) = l, this consists of l distinct points, in finite position. It is also straightforward to check that deg(L(W′)/L(x)) = l. As the extension L(W′)/L(x) is Galois, Card(G) = l. If the action fails to be transitive, then, by a simple counting argument, we can find a non-trivial g_{0} ∈ G and x_{0} ∈ W′(x), with g_{0} ▪ x_{0} = θ_{g_{0}}(x_{0}) = x_{0}. As the morphisms θ_{g_{0}} and θ_{Id} = Id are etale, it follows that θ_{g_{0}} = Id, see Proposition 3.15 of [17], which is a contradiction. Sharp transitivity is obtained by a similar argument. This proves (i).

By Lemma 2.5, we have that the power series {η_{1}(x), …, η_{m}(x)} constitute a complete set of distinct roots for the polynomial F (x, y), considered as belonging to L(x)[y]. By (†) of Lemma 3.3, and the later remark there, the power series {η_{1}(x), …, η_{m}(x)} also belong to the coordinate ring R(V′) ⊂ L(C′). In particular, we have that the function field L(C′) is a splitting field for the polynomial F (x, y), over the subfield L(x), and G acts transitively on the power series {η_{1}(x), …, η_{m}(x)} ⊂ R(V′) ⊂ R(W′), (♯). We can naturally extend the action of G to the coordinate ring R(W) = R(W′)[y]. By (♯), this defines a transitive action on the irreducible curves {D′_{1}, …, D′_{m}}, which extends to an action on the flashes {C_{1}, …, C_{m}}, using elementary facts on Zariski closure and the definition of C_{j} = \overline{D′_{j}}, for 1 ≤ j ≤ m. (This action clearly also respects the equivalence relation, defined in footnote 9.) Hence, (ii) is shown. Algebraic definability of the group action follows easily, by enumerating the parameters involved in the action defined by (∗), and restricting to the variety W′. □

We now give a more refined version of Lemma 3.3, by analysing the intersections and singularities of the flashes {C_{j} : 1 ≤ j ≤ m}.

We first observe the following, we recall the notion of val_{γ}, for a birational morphism, from [7].

Lemma 3.24. Let C′ satisfy the conditions of Lemmas 3.3 and 3.21, then, if a branch γ of C′ is centred in finite position along the axis x = 0, the values of the birational morphisms val_{γ}(η_{j}) are distinct, belonging to the set {a_{1}, …, a_{m}}.

Proof. Suppose that γ is a branch, centred along x = 0. If b_{j} denotes the value val_{γ}(η_{j}), and γ is centred at (0, c), then (0, b_{j}, c) belongs to the curve C_{j}. By (ii)(a) of Lemma 3.21, the projection (0, b_{j}) belongs to C, hence, b_{j} belongs to the set {a_{1}, …, a_{m}}. Suppose that val_{γ}(η_{j_{0}}) = val_{γ}(η_{j_{1}}) = a_{j}, for j_{0} ≠ j_{1}. The birational morphisms {η_{j_{0}}, η_{j_{1}}} are distinct, by Lemma 3.21, hence, there exists an open subset U ⊂ C′, for which (η_{j_{0}} − η_{j_{1}}) is non-zero. Let (x, z) ∈ (γ ∩ U), then the values η_{j_{0}}(x, z) and η_{j_{1}}(x, z) are distinct and the points {(x, η_{j_{0}}(x, z), z), (x, η_{j_{1}}(x, z), z)} belong to the curves (C_{j_{0}} ∩ V_{(0,a_{j},c)}) and (C_{j_{1}} ∩ V_{(0,a_{j},c)}) respectively. Applying Lemma 3.21, the projected points {(x, η_{j_{0}}(x, z)), (x, η_{j_{0}}(x, z))} belong to C ∩ V_{(0,a_{j})}. As x ∈ V_{0}, it follows that I_{(0,a_{j})}(C, x = 0) ≥ 2. This clearly contradicts the presentation of C, given in Lemma 1.1(iii).

□

We recall that C is a nodal curve, satisfying the assumptions of Lemma 1.1.

Definition 3.25. Let ℘ = {ρ_{1}, …, ρ_{i}, …, ρ_{r}} enumerate the branches of C with the following properties, relative to the coordinate system (x, y);

Either (i). The tangent line of ρ_{i} to C is vertical.

Or (ii). ρ_{i} is centred at a node, belonging to C.

By Lemma 1.1, all such branches are in finite position, and the conditions are mutually exclusive.

In the terminology of Lemma 3.21, we observe that the projection pr_{1} defines a quasi finite map from (C_{j} \ {Q_{1}}) to C, which extends to give a well-defined map from the branches γ of C_{j} to C, and that the branches of C_{j} can be naturally identified with the branches of C′, using the birational map η_{j}. This motivates the following definition;

Definition 3.26. For the branches in Definition 3.25, we define, Γ_{j} = {γ_{1,j}, …, γ_{t,j}} to be the lifting of these branches to the flash C_{j}, in Lemma 3.21, and also of the projective curve C′, given in Lemma 3.24. We define Γ = {γ_{1}, …, γ_{t}} = ⋃_{1≤j≤m} Γ_{j}.

We have the following birational invariance;

Lemma 3.27. Let α_{12} : C^{1} ↭ C^{2} satisfy the conditions of Lemma 3.3 and Lemma 3.16, with equivalent flashes θ_{j} : C_{j}^{1} ↭ C_{j}^{2}, defined by Lemma 3.21, see also footnote 9. Then the branches {γ, η_{j}(γ)} of {C^{1}, C_{j}^{1}} lift the same branch ρ of C, as {α_{12}(γ), θ_{j}(η_{j}(γ))}. In particular, the sets Γ_{j} and Γ are birationally invariant, and val_{γ}(η_{j}) = val_{α_{12}(γ)}(η_{j}).

Proof. We denote the representatives of the power series η_{j} by the rational functions f ∈ L(C^{1}) and g ∈ L(C^{2}). By the definition of α_{12}, we have that (g ∘ α_{12}) = f, and (pr_{1} ∘ α_{12}) = pr_{1}, as rational functions on C^{1}. It is sufficient to show that (pr_{1} ∘ η_{j}) = (pr_{1} ∘ η_{j} ∘ α_{12}) : C^{1} → C, as rational maps. For a generic point (x, z) ∈ C^{1}, we have that;

(pr_{1} ∘ η_{j})(x, z) = pr_{1}(x, f(x, z), z) = (x, f(x, z))

(pr_{1} ∘ η_{j} ∘ α_{12})(x, z) = (pr_{1} ∘ η_{j})(x, (pr ∘ α_{12})(x, z))

= pr_{1}(x, g(x, (pr ∘ α_{12})(x, z)), z)

= pr_{1}(x, (g ∘ α_{12})(x, z), z)
= (x, f(x, z)) (^{10})

Hence, the result is shown. □

We make the following definition;

Definition 3.28. Let C′ satisfy the conditions of Lemmas 3.3 and 3.21, then, we define a branch to be geometric, if it is either finite, or, hyperbolic and centred at Q_{1}. We define a branch to be perpendicular, if it is infinite, and also parabolic, if centred at Q_{1}. See Definition 2.1 for relevant terminology.

Remarks 3.29. A branch is geometric if and only if it is not perpendicular. This follows immediately from Definition 2.1. The notions of geometric and perpendicular are also birationally invariant, in the sense of Lemma 3.16.

Lemma 3.30. Let C′ satisfy the conditions of Lemmas 3.3 and 3.21, then, for a geometric branch γ of C′, we have that; val_{γ}(η_{j_{0}}) = val_{γ}(η_{j_{1}}), for j_{0} ≠ j_{1}, iff γ ∈ (Γ_{j_{0}} ∩ Γ_{j_{1}})

10We have temporarily used the notation pr for the projection onto the zcoordinate.

There does not exist a geometric branch γ of C′, on which {η_{j_{0}}, η_{j_{1}}, η_{j_{2}}} take the same val_{γ}, for a distinct triple (j_{0}, j_{1}, j_{2})

Proof. By Lemma 3.10 and Remarks 3.29, we can assume that γ is in finite position. If val_{γ}(η_{j}) = ∞, then the point [0 : 1 : 0 : 0] belongs to the flash C_{j}, hence, the point [0 : 1 : 0] belongs to the curve C. This contradicts the presentation of C in Lemma 1.1(iii). Suppose γ is centred at (x_{0}, z_{0}), with val_{γ}(η_{j_{0}}) = val_{γ}(η_{j_{1}}) = y_{0} < ∞. Then the point (x_{0}, y_{0}, z_{0}) belongs to the intersection C_{j_{0}} ∩ C_{j_{1}}, and projects to a point (x_{0}, y_{0}) on C. We can choose (x, z) ∈ V_{(x_{0},z_{0})}, such that η_{j_{0}}(x, z) ≠ η_{j_{1}}(x, z), and, therefore, find distinct points {y, y′}, with (x, y, z) ∈ (V_{(x_{0},y_{0},z_{0})} ∩ C_{j_{0}}) and (x, y′, z) ∈ (V_{(x_{0},y_{0},z_{0})} ∩ C_{j_{1}}). The distinct projected points (x, y) and (x, y′) belong to (V_{(x_{0},y_{0})} ∩ C). It follows that I_{(x_{0},y_{0})}(C, x = x_{0}) ≥ 2. By the presentation of C in Lemma 1.1, this can only occur if the branch(es), centred at (x_{0}, y_{0}), belong to ℘. It follows, from Definition 3.26, that the branch γ ∈ (Γ_{j_{0}} ∩ Γ_{j_{1}}). Conversely, suppose that γ ∈ (Γ_{j_{0}} ∩ Γ_{j_{1}}), so the projected branches pr_{1}(η_{j_{0}}(γ)) and pr_{1}(η_{j_{1}}(γ)), (∗), belongs to ℘. Supposing that val_{γ}(η_{j_{0}}) ≠ val_{γ}(η_{j_{1}}), then the projections, (∗), are centred at distinct points (x_{0}, y_{0}), (x_{0}, y_{1}) of C. This contradicts Lemma 1.1(iv). Finally, suppose that val_{γ}(η_{j_{0}}) = val_{γ}(η_{j_{1}}) = val_{γ}(η_{j_{2}}) = y_{0} < ∞. By a similar argument to the above, we would find a point (x_{0}, y_{0}) ∈ C, with I_{(x_{0},y_{0})}(C, x = x_{0}) ≥ 3. This contradicts Lemma 1.1(ii).

□

As a consequence of the preceding lemma, we obtain;

Lemma 3.31. Let C′ satisfy the conditions of Lemma 3.30, then we have that;

Γ = ⨆_{1≤j_{0}<j_{1}≤m}(Γ_{j_{0}} ∩ Γ_{j_{1}})

is a disjoint union of pairwise intersections.

Proof. Suppose that γ ∈ Γ, therefore, by Definition 3.26, belongs to Γ_{j_{0}}, for some 1 ≤ j_{0} ≤ m. We claim that γ is geometric. For suppose that γ is perpendicular, then, by Lemma 3.10, it can be centred at [1 : 0 : 0]. One can verify, that the projected branch pr_{1}(η_{j_{0}}(γ)) is centred on the line l_{∞}, (^{11}). However, (℘ ∩ l_{∞}) = ∅, by Lemma 1.1(i),(ii). By Lemma 3.10 and Remarks 3.29, the branch γ can be centred in finite position, (x_{0}, z_{0}), and the projected branch pr_{1}(η_{j_{0}}(γ)) is centred, in finite position, along the axis x = x_{0}. Suppose that the values val_{γ}(η_{j}), (∗), are distinct from val_{γ}(η_{j_{0}}), for j ≠ j_{0}. If these values, (∗), are, themselves, different, then, using the observation that the values are finite, we clearly obtain m seperate points of C, along the axis x = x_{0}. However, the projected branch ρ_{0} = pr_{1}(η_{j_{0}}(γ)) belongs to ℘, and, therefore, I_{ρ_{0}}(C, x = x_{0}) = 2. It follows, by Bezout’s theorem, that there can only exist m − 1 different points along the axis x = x_{0}. It follows that there must exist {η_{j_{1}}, η_{j_{2}}}, with (j_{0}, j_{1}, j_{2}) a distinct triple, such that val_{γ}(η_{j_{1}}) = val_{γ}(η_{j_{2}}). Now, by Lemma 3.30, the projected branch ρ_{1} = pr_{1}(η_{j_{1}}(γ)) belongs to ℘. It follows that we obtain two different branches {ρ_{0}, ρ_{1}} ⊂ ℘, centred at seperate points along the axis x = x_{0}. This contradicts Lemma 1.1(iv). Hence, there must exist j_{0} ≠ j_{1}, with val_{γ}(η_{j_{0}}) = val_{γ}(η_{j_{1}}). In particular, pr_{1}(η_{j_{1}}(γ)) belongs to ℘. Therefore, γ ∈ (Γ_{j_{0}} ∩ Γ_{j_{1}}). If the intersection of two of the sets in ⋃_{1≤j_{0}<j_{1}≤m}(Γ_{j_{0}} ∩ Γ_{j_{1}}), (∗∗), were non-empty, then, using the first part of Lemma 3.30, we would obtain a distinct triple (j_{0}, j_{1}, j_{2}), with val_{γ}(η_{j_{0}}) = val_{γ}(η_{j_{1}}) = val_{γ}(η_{j_{2}}). This contradicts the second part of Lemma 3.30. Hence, the union (∗∗) is disjoint. □

We make the following definition;

Definition 3.32. Let γ be a blue branch of C′, in finite position, centred at (x_{0}, z_{0}). We define a function h ∈ L(C′) to be symmetric at γ, if;

For generic x_{1} ∈ V_{x_{0}}, and {(x_{1}, y_{1}), …, (x_{1}, y_{t})} = (γ ∩ x = x_{1}), we have that;

h(x_{1}, y_{1}) = … = h(x_{1}, y_{t}), (∗).

Remarks 3.33. This is a good definition. Without loss of generality, assume that val_{γ}(h) = y_{0}, and let γ′ be the corresponding branch, centred at (x_{0}, y_{0}, z_{0}), of the image curve C′_{h} ⊂ P^{3}, with h : C′ ↭ C′_{h}, and γ′′ the corresponding branch, centred at (x_{0}, y_{0}), of the projected curve pr_{1}(C′_{h}), with pr_{1} : C′_{h} ⇝ pr_{1}(C′_{h}). We are then able to observe that Card(γ′′ ∩ (x = x_{1})) = 1, witnessed by the projections;

11The case that val_{γ}(η_{j_{0}}) is infinite, being the more difficult calculation. We approximate the point [1 : 0 : 0], by [\frac{1}{ϵ} : 1 : 1] and obtain the approximation [\frac{1}{ϵ} :

\frac{1}{ϵ^{j}} u(ϵ) : 1 : 1] for the centre of the branch η_{j_{0}}(γ). This specialises to [1 : u(0) : 0 : 0] and projects to [1 : u(0) : 0], on l_{∞}. See similar calculations, previously in the paper.

(x_{1}, h(x_{1}, y_{1})) = … = (x_{1}, h(x_{1}, y_{t}))

It follows that I_{γ′′}(pr_{1}(C′_{h}), x = x_{0}) = 1, by definitions in [6]. In particular, the branch γ′′ is non-singular, and cuts the axis x = x_{0} transversely. By reversing the argument, we deduce easily that, for any x_{1} ∈ V_{x_{0}}, the property (∗) of Definition 3.32 holds.

We observe the following symmetry property;

Lemma 3.34. Let C′ satisfy the above conditions and let γ be a blue branch of C′ in finite position. Then, it is not the case that every function {η_{1}(x), …, η_{m}(x)} is symmetric at γ.

Proof. Suppose, for contradiction, that each function {η_{1}(x), …, η_{m}(x)} is symmetric at γ. Let γ be centred at (x_{0}, y_{0}), and choose x_{1} ∈ V_{x_{0}} generically, with {(x_{1}, y_{1}), …, (x_{1}, y_{t})} = (γ ∩(x = x_{1})). As the branch γ is blue, we have that t ≥ 2. By Lemma 3.23(i), we can find a birational morphism θ_{g} : C′ ↭ C′, with θ_{g}(x_{1}, y_{1}) = (x_{1}, y_{2}). θ_{g}^{∗} defines an isomorphism of the function field L(C′)/L(x), in particular it permutes the set of roots {η_{1}(x), …, η_{m}(x)}. The permutation is non-trivial, as L(C′) is generated by these roots, over L(x), and θ_{g}^{∗} ≠ Id. Without loss of generality, we can, therefore, assume that θ_{g}^{∗}(η_{1}) = η_{2}. As both (x_{1}, y_{1}) and θ_{g}(x_{1}, y_{1}) belong to γ, we must have that γ is invariant under the action of θ_{g}. In particular, for any x_{1} ∈ V_{x_{0}}, with x_{1} ≠ x_{0}, θ_{g} defines a non trivial permutation of the fibre γ ∩ (x = x_{1}) Now, for any point (x_{1}, y_{1}) on the branch γ, distinct from (x_{0}, y_{0}), using the assumption of symmetry, we have that;

η_{2}(x_{1}, y_{1}) = θ_{g}^{∗}(η_{1})(x_{1}, y_{1}) = η_{1}(θ_{g}(x_{1}, y_{1})) = η_{1}(x_{1}, y_{1})

This implies that the functions η_{1} and η_{2} are identical, which is not the case. Hence, one of the functions {η_{1}(x), …, η_{m}(x)} is not symmetric at γ.

□

Remarks 3.35. Definition 3.32 can be easily extended to include the case of a blue branch centred at [1 : 0 : 0]. Namely, we define a function h ∈ L(C′) to be symmetric at γ, if (∗) holds, with the requirement that x_{1} ∈ V_{∞} is generic, (in which case the intersections γ ∩ x = x_{1} are all in finite position). It is a straightforward exercise, left to the reader, to check that Remarks 3.33, and Lemma 3.34 hold for this definition.

We can now specify the positions of the blue and silver branches of C′;

Lemma 3.36. Suppose that the r vertical tangents of C are situated along the axes {x = a_{1}, …, x = a_{r}}, in the coordinate system (x, y), then the geometric blue branches of C′, up to birationality, see Definitions 3.11 and 3.28, are situated exactly along the same axes, in the coordinate system (x, z), and the geometric silver branches of C′, up to birationality, are situated exactly along the complementary axes, that is axes of the form x = b, for b distinct from {a_{1}, …, a_{r}}. In particular, the geometric branches, which, up to birationality, are situated along the line x = 0 are silver.

Proof. Suppose, for contradiction, that a blue branch γ, in finite position, is situated along a complementary axis x = b. By Lemma 3.34, we can assume that the function η_{1}(x) is not symmetric at γ. As we saw in Lemma 3.30, val_{γ}(η_{1}) = d < ∞. We, therefore, obtain, by the standard method, a corresponding branch γ′ of C, centred at (b, d). As η_{1} is not symmetric, it is a straightforward calculation, to show that I_{γ′}(C, x = b) ≥ 2. This implies that γ′ is a vertical tangent of C, contradicting the supposition of the lemma. Any geometric blue branch can be centred in finite position, using Lemma 3.10 and Definition 3.11. If γ′ defines a vertical tangent of C, centred along x = a_{j}, for a given 1 ≤ j ≤ r, then, using Lemma 3.30, we can find a corresponding branch γ, up to birationality, in finite position, along the same axis x = a_{j}. It is a simple calculation, to show that γ must be a blue branch. This show the first part of the lemma. The second part follows immediately from Lemma 3.13. The final part follows from Lemma 1.1(iii). □

Lemma 3.37. All the perpendicular branches of C′ are silver.

Proof. Suppose, for contradiction, that γ is a perpendicular blue branch. By Definition 3.28, Lemma 3.10 and Definition 3.11, we can assume that γ is centred at [1 : 0 : 0]. As I_{γ}(C′, l_{∞}) ≥ 2, by Definition 3.11, and, without loss of generality, the function η_{1}(x) is not symmetric at γ, by Remarks 3.35, we can find distinct points [\frac{1}{ϵ} : z_{1} : 1] and [\frac{1}{ϵ} : z_{2} : 1], belonging to γ ∩ x = \frac{1}{ϵ}, with η_{1}(\frac{1}{ϵ}, z_{1}) ≠ η_{1}(\frac{1}{ϵ}, z_{2}). The distinct points \{[\frac{1}{ϵ} : η_{1}(\frac{1}{ϵ}, z_{1}),: 1], [\frac{1}{ϵ} : η_{1}(\frac{1}{ϵ}, z_{1}),: 1]\} belong to the projected image branch γ′ of C′, centred along the line l_{∞}. It follows that I_{γ′}(C′, l_{∞}) ≥ 2. This contradicts Lemma 1.1(i). □

We can use the above specification to simplify the analysis of the nodes of C. We make the following refinement of terminology;

Definition 3.38. Let ℘_{ν} enumerate the branches of C, satisfying Definition 3.25(ii), Γ_{j,ν} the lifting of these branches to C_{j} and C′, as in Definition 3.26, and Γ_{ν} = ⋃_{1≤j≤m} Γ_{j,ν}.

It is a straightforward exercise to refine Lemma 3.27 and Lemma 3.31, namely;

Lemma 3.39. The sets Γ_{j,ν} and Γ_{ν} are birationally invariant, and Γ_{ν} = ⨆_{1≤j_{0}<j_{1}≤m}(Γ_{j_{0},ν} ∩ Γ_{j_{1},ν}) is a disjoint union of pairwise intersections.

Definition 3.40. If γ ∈ Γ_{ν}, we say that γ is associated to a node ν_{0} of C, if, pr_{1}(η_{j_{0}}(γ)) and pr_{1}(η_{j_{1}}(γ)), determined uniquely by Lemma 3.39, form the branches of ν_{0}.

We can define an equivalence relation on Γ_{ν} as follows;

Definition 3.41. Let γ_{1} and γ_{2} belong to Γ_{ν}. We define γ_{1} ∼ γ_{2} if they are associated to the same node of C.

Remarks 3.42. It is an interesting exercise to show that, if the s nodes of C are situated along the axes {x = b_{1}, …, x = b_{s}}, in the coordinate system (x, y), then any 2 branches of C′, which, up to birationality, can be centred along a given axis x = b_{j}, for some 1 ≤ j ≤ s, in the coordinate system (x, z), belong to Γ_{ν} and are equivalent. It is also clear, from the above, that the equivalence relation on Γ_{ν} has exactly s classes.

With these definitions, we have;

Lemma 3.43. There exists C′, satisfying the conditions of Lemma 3.3 and Lemma 3.21, with s branches {γ_{1}, …, γ_{s}} belonging to Γ_{ν}, representing each equivalence classes in Definition 3.41, such that each branch is isolated and in finite position.

Proof. By Lemma 3.36, and Lemma 1.1(iv), all the branches belonging to Γ_{ν} are silver. We can represent each equivalence class by s distinct silver branches, which, up to birationality, can be centred along the distinct coordinate axes {x = b_{1}, …, x = b_{s}}. By Lemma 3.14, we can simultaneously isolate these branches in finite position. □

The following result is geometrically clarifying;

Lemma 3.44. Flash Intersections

Let hypotheses and notation be as in the preceding lemmas, then;

(i). All the branches of the flashes {C_{1}, …, C_{m}} are non-singular.

(ii). No three flashes intersect in a point. That is there is no point p ∈ S and distinct flashes {C_{j_{1}}, C_{j_{2}}, C_{j_{3}}}, with j_{1} < j_{2} < j_{3}, such that p ∈ (C_{j_{1}} ∩ C_{j_{2}} ∩ C_{j_{3}}).

(iii). There exists a mapping from the intersections of flashes to the vertical tangent points and nodes of C. That is;

a. For every point p ∈ S and distinct flashes {C_{j_{1}}, C_{j_{2}}}, with j_{1} < j_{2}, such that p ∈ (C_{j_{1}} ∩ C_{j_{2}}), we have that pr_{1}(p) defines a vertical tangent point or a node of C in the sense of Lemma 1.1(ii).

b. For every p_{1} ∈ C, defining a vertical tangent point or a node, there exists a p ∈ S and distinct flashes {C_{j_{1}}, C_{j_{2}}}, with j_{1} < j_{2}, such that p ∈ (C_{j_{1}} ∩ C_{j_{2}}) and pr_{1}(p) = p_{1}.

(iv). There exists a bijection between the intersections of flashes and the spirit lines. That is;

a. Every intersection between two distinct flashes {C_{j_{1}}, C_{j_{2}}}, with j_{1} < j_{2}, lies on a unique spirit line l ⊂ S.

b. For every spirit line l ⊂ S, there exists a unique point p ∈ S and distinct flashes {C_{j_{1}}, C_{j_{2}}}, with j_{1} < j_{2}, such that p ∈ (l ∩ C_{j_{1}} ∩ C_{j_{2}}).

(v). In particular, there exists a mapping from the spirit lines to the vertical tangent lines of C.

Proof. We adopt the coordinate of Definition 3.1. The coordinate projection pr_{1} defines a quasi finite map from (C_{i} \ {Q_{1}}) to C, which extends to give a well-defined map from the branches γ of C_{i} to C. This follows from the observation that there exists a unique morphism ϕ from the desingularisation C_{i}^{ns} to C^{ns}, for which (pr_{1} ∘ θ_{i}) = (θ ∘ ϕ), where {θ, θ_{i}} are the desingularization morphisms of {C, C_{i}} respectively. We claim that if γ is a singular branch of C_{i}, then pr_{1}(γ) is a singular branch of C, (∗). Let pr_{1}(γ) be centred at A ∈ P^{2}, and let γ be centred at B ∈ P^{3}. Let l be a line passing through A. Let P_{l} be the plane spanned by the lines {l, l_{AQ_{1}}}, which must pass through the point B. If γ is singular, then I_{italian,γ}(C_{i}, P_{l}) ≥ 2, hence, we can choose a plane P′_{l}, infinitely close to P_{l}, with C_{i}∩P′_{l}∩V_{B}, containing two distinct points {B_{1}, B_{2}}. By an appropriate choice of P′_{l}, we can assume that B_{2} ∉ l_{B_{1}Q_{1}}. Let l′ = P^{2}∩P′_{l}, then {pr_{1}(B_{1}), pr_{1}(B_{2})} are distinct points belonging to C ∩ l′ ∩ V_{A}. It follows that pr_{1}(γ) must be singular, showing (∗). By the hypothesis that C is non-singular, it follows that all the branches of C_{i} are non-singular, giving (i). Suppose, for contradiction, that three flashes intersect in a point p ∈ S. Without loss of generality, assume that q = pr_{2}(p) ∈ C′ is in finite position. Choosing generic q′ ∈ V_{q}, we can assume that the functions {η_{j_{1}}, η_{j_{2}}, η_{j_{3}}} are defined at q′, with {η_{j_{1}}(q′), η_{j_{2}}(q′), η_{j_{3}}(q′)} ⊂ V_{p}. As the flashes {C_{j_{1}}, C_{j_{2}}, C_{j_{3}}} are distinct, and therefore intersect in finitely many points, by assuming q′ is generic, we must have that the y-coordinates {y_{j_{l}}, y_{j_{2}}, y_{j_{3}}} of the triple {η_{j_{1}}(q′), η_{j_{2}}(q′), η_{j_{3}}(q′)} are also distinct, (†). Suppose that pr_{1}(p) ∈ C has coordinates (a, b) in the system (x, y), then, the coordinates of {pr_{1}(η_{j_{1}}(q′)), pr_{1}(η_{j_{2}}(q′)), pr_{1}(η_{j_{3}}(q′))} ⊂ C are of the form {(a′y_{j_{l}}), (a′, y_{j_{2}}), (a′, y_{j_{3}})}, with a′ ∈ V_{a}, and {y_{j_{l}}, y_{j_{2}}, y_{j_{3}}} ⊂ V_{b}. It follows that I_{(a,b)}(C, x = a) = 3, contradicting non-singularity of C or Lemma 1.1(ii). This shows (ii). The remaining cases (iii) and (iv) are left as an exercise for the reader, It is mainly a question of converting the results of Lemmas 3.30 and Lemma 3.31. One should be careful, to exclude exceptional cases, arising from the images of perpendicular branches.

□

4. Asymptotic Degenerations

The flashes model of a non-singular algebraic curve C is useful for studying the problem of degenerations of curves to lines in general position, as, such degenerations can be viewed as a continuous family of flashes, (♯). (^{12}) This is made clearer by the following;

Definition 4.1. Asymptotic Degenerations

12The modern catastrophe theory approach to the problem seems, at least from an aesthetic and practical point of view, to have certain deficiencies.

Let {C_{t}}_{t∈P^{1}} be a family of plane projective algebraic curves of degree m ≥ 2, with the following properties, relative to a fixed coordinate system {x, y};

(i). The intersection of each curve C_{t} in the family with the axis x = 0 consists of a fixed set of m distinct points {(0, a_{1}), …, (0, a_{j}), …, (0, a_{m})} in finite position, distinct from (0, 0). (^{13}).

(ii). For generic t ∈ P^{1}, C_{t} is irreducible.

(iii). Each curve C_{t} has m distinct non-singular branches, centred at the points {(0, a_{1}), …, (0, a_{m})}, and, the tangent lines {l_{a_{1}}, …, l_{a_{m}}} at these branches are all fixed and distinct from the axis x = 0.

We call such a family of curves an asymptotic degeneration.

Lemma 4.2. Existence of Asymptotic Degenerations

Any nodal curve C is part of a non-trivial asymptotic degeneration.

Proof. Given a plane nodal curve C, the existence of asymptotic degeneration of C, follows from;

(i). Lemma 1.1. (ii). Dimension calculations for the space of such curves, with fixed degree and number of nodes.

The reader should look at [19] and [28] for more details about the calculations in (ii). Any such curve C is part of an irreducible space of dimension 3m − 1 + k, where m = deg(C), and k is a non-negative integer depending on the number of nodes. The condition on asymptotes imposes at most m + 1 ≤ 3m − 1 ≤ 3m − 1 + k conditions on such a space, (∗). This follows from the observation that the condition on tangent lines in 3.28(iii) is a consequence, (∗∗), of 3.28(i), 3.28(ii), Lemma 1.1 and the extra single condition that the tangent line to one branch, centred along the axis x = 0 is fixed. The proof of (∗∗) is straightforward, suppose that we have a family of curves {C_{t}}_{t∈P^{1}}, satisfying the conditions 3.28(i),(ii), with the branches of the generic curve, centred at {(0, a_{1}), …, (0, a_{m})}, transverse to x = 0. Using an analytic representation of a given branch, at (0, a_{j}), see [6], the gradient of the tangent line at (0, a_{j}) is given by the rational function θ_{j}(t) = −\frac{f_{x}(x,y;t)}{f_{y}(x,y;t)}|_{(0,a_{j})}. If the condition on tangent lines in 3.28(iii) fails, then θ_{j_{0}}(t) is non-constant, for some 1 ≤ j_{0} ≤ m. In this case, by compactness of the parameter space P^{1}, it attains the value ∞, for finitely many {t_{1}, …, t_{r}}. This implies that f_{y}(x, y; t_{j}) = 0, for 1 ≤ j ≤ r, and the branch, at (0, a_{j_{0}}) is either tangent to the axis x = 0, or singular for the curve C_{t_{j}}. By Bezout’s theorem, it follows that C_{t_{j}} either has at most m − 1 distinct points of intersection with the axis x = 0, contradicting the condition 3.28(i), or contains the axis x = 0 as a component. This last case is excluded by the condition that the tangent line to one branch, centred along the axis x = 0 is fixed. The stipulation that the curve does not contain the point (0, 0) is simply achieved by a translation.(^{14})

13It is, on occasion, useful to impose the following additional assumptions;

(a). Using projective coordinates {X, Y, W }, the intersection of each curve C_{t} in the family, with the axis W = 0, consists of a fixed set of m different points, distinct from [1 : 0 : 0].

(b). The tangent line to one of the branches, centred at a point in (b), is fixed, and distinct from the axis W = 0.

This is technically convenient in terms of later terminology, and is required to exclude extremely exceptional cases, which will be mentioned later.

□

Lemma 4.3. Uniformity of Asymptotic Degenerations

Let {C_{t}}_{t∈P^{1}} be an asymptotic degeneration, of fixed degree, then, there exists a corresponding algebraic family of plane curves, {D_{r} : r ∈ R}, in a fixed coordinate system {x, z} with R → P^{1} a finite cover, such that;

(i). For generic r ∈ R, D_{r} is irreducible, (0, 0) ∈ D_{r}, and (D_{r}, (0, 0)) is an etale cover of (A^{1}, (0, 0)).

(ii). For non-generic r_{∞} ∈ R, corresponding to t_{∞}, with C_{t_{∞}} having components {C_{t_{∞}}^{1}, …, C_{t_{∞}}^{k}}, D_{r_{∞}} is a union of component curves {D_{r_{∞}}^{1}, …, D_{r_{∞}}^{l}}, with l ≥ k.

There exists an algebraic family of parameters, s ∈ S, with S → R → P^{1} a finite cover, and a sequence of morphisms {η_{1,s}, …, η_{j,s}, …, η_{m,s} : s ∈ S}, with the properties;

14To obtain the extra exceptional requirements in footnote 11, one needs to impose a further m + 1 conditions. Lemma 1.1 allows one to find two distinct axes, which cut the given curve transversely. The existence of such strongly asymptotic degenerations follows again from the fact that 2(m + 1) ≤ 3m − 1 ≤ 3m − 1 + k, if m ≥ 2. The case m = 1 is just a line.

(iii). For corresponding generic s ∈ S, r ∈ R, t ∈ P^{1};

(a). η_{j,s}|D_{r} is birational, of fixed degree, and η_{j,s}(0, 0) = (0, a_{j}, 0), relative to the coordinate systems (x, z) ⊂ (x, y, z).

(b). The sequence of image curves {C_{1,r,s}, …, C_{j,r,s}, …, C_{m,r,s}} are distinct irreducible covers of C_{t}, using the projection pr_{1} from the coordinate system {x, y, z} to {x, y}.

(iv). For corresponding non-generic (s_{∞}, r_{∞}, ∞) ∈ S × R × P^{1}, the reduced, (^{15}), functions {η_{1,s_{∞}}, …, η_{j,s_{∞}}, …, η_{m,s_{∞}}} restrict to morphisms on each component {D_{r_{∞}}^{1}, …, D_{r_{∞}}^{i}, …, D_{r_{∞}}^{t}} of D_{r_{∞}}, with the properties that;

(a). η_{j,s_{∞}}|D_{r_{∞}}^{i} is birational.

(b). Each image curve C_{i,j,∞} is a cover of an irreducible component of C_{∞}.

(c). The intersections (C_{i_{0},j_{0},∞} ∩ C_{i_{1},j_{1},∞}) are finite for (i_{0}, j_{0}) ≠ (i_{1}, j_{1}).

(d). For any given j, every irreducible component of C_{∞} is covered by an image curve C_{i,j,∞}.

(v). For 1 ≤ j ≤ m, the family of image curves {C_{j,r,s}}, including the non-generic limit curves {C_{i,j,∞}}, form an irreducible closed variety over a 1-parameter space S_{j}, with S → S_{j} → R → P^{1} finite covers.

Proof. Let P^{N} denote the parameter space for plane curves of degree n, in the coordinate system \{X, Z, W \}, where N = \frac{n(n+3)}{2}. We, first, observe that there exists a constructible set V ⊂ P^{N}, with the properties that, for \overline{r} ∈ V;

(i). The corresponding curve D\overline{r} is irreducible.

15By which we mean the functions contain no factors, vanishing or taking the value ∞, entirely on a component of D_{r_{∞}}; removing such factors may lower their degree.

54 TRISTRAM DE PIRO

(ii). D\overline{r} passes through (0, 0), in the coordinate system (x, z), where \{x = \frac{X}{W} , z = \frac{Z}{W} \}.

(iii). (D\overline{r}, (0, 0)) defines an etale cover of (A^{1}, 0). (∗) Let F_{n}(X, Z, W, \overline{r}) = ∑_{i+j+k=n} r_{ijk}X^{i}Y^{j}W^{k} enumerate all plane curves

of degree n, and let F_{p} be similar polynomials, enumerating all plane curves of degree 1 ≤ p < n. Let

Φ_{1}(\overline{r}) ≡ ¬∃_{1≤p<n}\overline{x}_{1} … \overline{x}_{p} … \overline{x}_{n−1}∃_{1≤q<n}\overline{y}_{1} … \overline{y}_{q} … \overline{y}_{n−1}

[⋁_{p+q=n} F_{n}(x, z, \overline{r}) = F_{p}(x, z, \overline{x}_{p})F_{q}(x, z, \overline{y}_{q})] By standard properties, Φ_{1} defines a constructible subset V_{1} ⊂ P^{N},

such that, (i) of (*), holds for \overline{r} ∈ V_{1}. Let Φ_{2}(\overline{r}) ≡ [Φ_{1}(\overline{r}) ∧ F_{n}(0, 0, \overline{r}) = 0] Then Φ_{2} defines a constructible set V_{2} ⊂ V_{1} ⊂ P^{N}, for which (ii) of

(*), holds as well. Let Φ(\overline{r}) ≡ [Φ_{2}(\overline{r}) ∧ \frac{∂F}{∂Z} (X, Z, W, \overline{r})|_{[0:0:1]} ≠ 0] It is an easy exercise, using the definition of an etale morphism, to see

that Φ defines a constructible set V ⊂ V_{2}, for which all the conditions of (*) are satisfied. We adopt the coordinate notation of Definition 3.1. Let P^{2N_{j}} be a parameter space for the set of rational functions of degree n_{j} on P^{2}, where N_{j} = \frac{n_{j}(n_{j}+3)}{2}. We observe, that, for a pair (t, \overline{r}) ∈ (P^{1} × V), there exists at most one rational function η_{j}(x, z), of fixed degree n_{j},(^{16}), with the properties that;

(i). η_{j}(x, z) is defined at (0, 0) and η_{j}(0, 0) = a_{j} (ii). The image curve C_{j,t,\overline{r}}, defined as the projective closure of; \{(x, y, z) : y = η_{j}(x, z), (x, z) ∈ D\overline{r}\} is a cover of C_{t}, using the projection pr_{1}, (**), (^{17})

16Having a uniquely determined parameter \overline{s_{j}} ∈ P^{2N_{j}}

For a pair (x, z) belonging to the infinitesimal neighborhood (V_{(0,0)} ∩ D\overline{r}), there can clearly exist at most one possible point (x, y, z), for which (x, y) belongs to (V_{(0,a_{j})} ∩ C_{t}), by the definition 3.28(iii). Hence, η_{j} is uniquely determined on this neighborhood, and, therefore, on the curve D\overline{r}, showing (**). By Lemma 3.21, for generic t ∈ P^{1}, (†), and appropriate \{n, n_{j}\}, there exists \overline{r} ∈ V and \overline{s_{j}} ∈ P^{2N_{i}}, such that (**) is satisfied for the triple (t, \overline{r}, \overline{s_{j}}). Let G_{n_{j}}(X, Z, W, \overline{z_{i}}) enumerate all rational functions of degree n_{j}, and let {r, s_{1}, …, s_{j}, …, s_{m}} denote new parameters, with each s_{j} abbreviating \overline{s_{j}}, and r abbreviating \overline{r}. Let

Ψ_{j}(t, r, s_{j}) ≡ [Φ(r) ∧ G_{n_{j}}(0, 0, s_{j}) = a_{j} ∧ ((F_{n}(x, z, r) = 0) →

C_{t}(x, G_{n_{j}}(x, z, s_{j})))]

and;

Ψ(t, r, s_{1}, …, s_{j}, … s_{m}) ≡ ∧_{1≤j≤m}Ψ_{j}(t, r, s_{j})

By elementary considerations, Ψ defines a constructible subset U of (P^{1} × V × P^{2N_{1}} × … P^{2N_{i}} × … P^{2N_{m}}). We let

Θ(t) ≡ ∃r∃s_{1} … ∃s_{j} … ∃s_{m}Ψ(t, r, s_{1}, …, s_{j}, …, s_{m}).

By (†), Θ defines an open subset of P^{1}. We define S = \overline{U} to be the closure of U in (P^{1} × V × P^{2N_{1}} × … P^{2N_{j}} × … P^{2N_{m}}), and also R ⊂ (P^{1} × \overline{V}) to be the projection pr(S). Without loss of generality, we can assume that S and R are irreducible, with the properties;

(i). pr : S → R → P^{1} is onto.

(ii). For a generic pair (t, r) ∈ R, there exists a unique corresponding triple (t, r, \overline{s}) ∈ S

(iii).For a generic pair (t, r) ∈ R, r ∈ V. (***)

17The conic projection on P^{3} is defined, excluding the exceptional point [0 : 0 :

1 : 0], in the case that a curve passes through this point, one still obtains a well defined projected curve, see the paper [6] for the explicit construction.

where \overline{s} = (s_{1}, …, s_{j}, …, s_{m}). This follows from elementary considerations on dimension, the above mentioned property of Θ, and (∗∗). We assume that dim(S) ≥ 2. We then make the simple observation that, for a suitable embedding of S in P^{L}, it is possible to find a hyperplane H ⊂ P^{L}, with the properties that;

(a). (H ∩ S) is irreducible and (dim(H) ∩ S) = dim(S) − 1.

(b). pr(H ∩ S) = P^{1}

(c). H passes through a generic triple (t, r, \overline{s}) ∈ S.

This follows from a generalised version of Bertini’s Theorem, see [14] or [15]. It follows, by induction, that we can obtain the statement (***), with the additional property that dim(S) = 1. We then rephrase the property (iii) of (∗);

(iii)’. For generic t ∈ P^{1}, the fibre R(t) ⊂ V.

The statements (i) and (iii)(a), (b) of the theorem now follow immediately from (***)(i),(ii),(iii), the distinctness claim following immediately from the fact that the functions η_{j,s} are different on the curve D_{r}, by (∗∗)(i). We now consider the non-generic cases in the theorem. We abbreviate the tuple \overline{s} to s and let;

S_{j}(t, r, s_{j}) ≡ ∃s_{1} … ∃s_{i≠j} … ∃s_{m}S(t, r, s_{1}, …, s_{i}, …, s_{m})

and U_{j} = pr_{j}(U), so that U_{j} ⊆ S_{j}. We consider the following varieties W_{j}, T_{j} ⊆ S_{j} × P^{3} defined by;

W_{j}(t, r, s_{j}, x, y, z) ≡ U_{j}(t, r, s_{j}) ∧ D_{r}(x, z) = 0 ∧ g_{s_{j}^{2}}(x, z) ≠ 0

∧ \frac{g_{s^{1}_{j}}(x,z)}{g_{s^{2}_{j}}(x,z)} = y

T_{j}(t, r, s_{j}, x, y, z) ≡ U_{j}(t, r, s_{j}) ∧ g_{s_{j}^{2}}(x, z) ≠ 0 ∧ \frac{g_{s^{1}_{j}}(x,z)}{g_{s^{2}_{j}}(x,z)} = y

where {g_{s_{j}^{1}}, g_{s_{j}^{2}}} are homogeneous polynomials, of degree n_{j}, represented by the parameter s_{j}. T_{j} is irreducible, as U_{j} × A^{2} is irreducible and a generic fibre T_{j}(t, r, s_{j}, x, z) consists of a single point. W_{j} is irreducible, as U_{j} is irreducible, and a generic fibre W_{j}(t, r, s_{j}) corresponds to the irreducible image curve C_{j,r,s} considered in case (iii)(b). It follows that the Zariski closures \overline{W_{j}}, \overline{T_{j}}, which we again denote by W_{j}, T_{j} ⊆ S_{j} × P^{3}, are also irreducible. We clearly have that W_{j} ⊂ T_{j}. It follows, by dimension considerations, that, for non-generic (t_{∞}, r_{∞}, s_{j,∞}), the fibre W_{j}(t_{∞}, r_{∞}, s_{j,∞}) defines a closed, possibly not irreducible, variety of dimension 1 C_{j,t_{∞},r_{∞}} ⊂ P^{3}, (^{18}). The aim of case (iv) in the theorem, is to describe these limit curves C_{j,t_{∞},r_{∞}} more explicitly, this will incidentally exclude the scenario of footnote 18.

We now examine the behaviour of the morphism η_{j,∞}(x, z), determined by the parameter s_{j,∞}, on the curve D_{t_{∞},r_{∞}}, and aim to show that the curve C_{j,t_{∞},r_{∞}} is obtained as the image of D_{t_{∞},r_{∞}}, after making certain reductions in the factorisation of η_{j,∞}, see footnote 15, (^{19}).

We consider the closed variety V_{j} ⊂ (S_{j} × P^{2}) defined by;

V_{j}(t, r, s_{j}, x, z) ≡ S_{j}(t, r, s_{j}) ∧ D_{r}(x, z)

V_{j} is equidimensional, as the space Q_{n} of degree n curves is irreducible, a proof is given in the paper [5], the variety (S_{j} × P^{2}) is irreducible, and V_{j} is obtained as the intersection of these varieties within a non-singular total projective space. Irreducibility follows easily from the fact that the generic curve D_{r} is irreducible, see the similar argument in the just quoted proof. Now suppose that (x_{0}, z_{0}) belongs to some component of the curve D_{t_{∞},r_{∞}}, then we can find s_{j,∞} with V_{j}(t_{∞}, r_{∞}, s_{j,∞}, x_{0}, z_{0}). By Lemma 3.2 of [4], we can find a generic (t, r, s_{j}, x_{1}, z_{1}), belonging to V_{j}, specialising to (t_{∞}, r_{∞}, s_{j,∞}, x_{0}, z_{0}). As dim(V_{j}) = 2, dim(x_{1}, z_{1}/t, r, s_{j}) = 1, in particular, we may assume that (x_{1}, z_{1}) does not belong to the curve defined by g_{s_{j}^{2}}(x, z) = 0, using also the fact that g_{s_{j}^{2}} cannot vanish identically on the irreducible curve D_{t,r}, by the description (iii)(a) of the theorem. It follows that we can find y_{1}, with W_{j}(t, r, s_{j}, x_{1}, y_{1}, z_{1}), and, by specialisation, there exists y_{0}, with W_{j}(t_{∞}, r_{∞}, s_{j,∞}, x_{0}, y_{0}, z_{0}). This shows that the ”projection” of the curve C_{j,t_{∞},r_{∞}} contains the curve D_{t_{∞},r_{∞}}, (****).

As the variety T_{j} is irreducible and W_{j} ⊂ T_{j}, by Lemma 3.2 of [4], if a point (x_{0}, y_{0}, z_{0}) lies on the limit curve C_{j,t_{∞},r_{∞}}, there exists a generic, (*****), (t, r, s_{j}, x_{1}, y_{1}, z_{1}) of T_{j}, specialising, (******), to (t_{∞}, r_{∞}, s_{j,∞}, x_{0}, y_{0}, z_{0}). We assume that the field < t_{∞}, r_{∞}, s_{j,∞}, x_{0}, y_{0}, z_{0} > is contained in the base field L. By (****), the facts that dim(S_{j}) = 1 and dim(T_{j}) = 3, we can assume that dim(t, r, s_{j}/L) = 1, and dim(x_{1}, z_{1}/L) = dim(x_{1}, z_{1}/L < t, r, s_{j} >) = 2. We pick independent infinitesimals {ϵ, δ_{1}, δ_{2}}, such that {t, r, s_{j}} ⊂ L(ϵ)^{alg}, {x_{1}, z_{1}} ⊂ L(δ_{1}, δ_{2})^{alg}, and the fields L(ϵ)^{alg}, L(δ_{1}, δ_{2})^{alg} are algebraically disjoint over L. By results of [3] or [4], we can construct a sequence of specialisations;

18It is conceivable that the variety could have extraneous points, that is irreducible components of dimension 0. We will refer to the variety as a curve in the course of the proof.

19Until (!), we make the assumption that D_{t_{∞},r_{∞}} does not contain the line W = 0 as a component in the projective coordinate system {X, Z, W }

π_{1} : P (L(ϵ, δ_{1}, δ_{2})^{alg}) = P (K(ϵ)^{alg}) → P (L(δ_{1}, δ_{2})^{alg}) = P (K)

π_{2} : P (L(δ_{1}, δ_{2})^{alg}) → P (L)

with π_{1} : P (L(ϵ)^{alg}) → P (L)

By the construction of a universal specialisation, see [3], we can take the composition π = π_{2} ∘ π_{1} to satisfy (******). By the definition of

T_{j}, we have that \frac{g_{s^{1}_{j}}(x_{1},z_{1})}{g_{s^{2}_{j}}(x_{1},z_{1})} = z_{1}, we wish to compute the specialisation

π(z_{1}). We first show that;

π_{1}(\frac{g_{s^{1}_{j}}(x_{1},z_{1})}{g_{s^{2}_{j}}(x_{1},z_{1})}) = π_{1}(\frac{g_{s{1 \atop j,∞}}(x_{1},z_{1})}{g_{s{2 \atop j,∞}}(x_{1},z_{1})}) (*******)

By a straightforward computation, abbreviating the values {g_{s_{j,∞}^{1}}(x_{1}, z_{1}), g_{s_{j,∞}^{2}}(x_{1}, z_{1})} by {k_{1}, k_{2}} ⊂ K, with k_{2} ≠ 0, we have;

\frac{g_{s^{1}_{j}}(x_{1},z_{1})}{g_{s^{2}_{j}}(x_{1},z_{1})} − \frac{g_{s{1 \atop j,∞}}(x_{1},z_{1})}{g_{s{2 \atop j,∞}}(x_{1},z_{1})} = \frac{k_{1}+ϵ_{1}}{k_{2}+ϵ_{2}} − \frac{k_{1}}{k_{2}} = \frac{k_{2}ϵ_{1}−k_{1}ϵ_{2}}{k_{2}^{2}+ϵ_{2}k_{2}}

where {ϵ_{1}, ϵ_{2}} are infinitesimals in K(ϵ)^{alg}. Clearly, the last term in the above expression specialises to zero, showing (*******). We now show that;

π_{2}(\frac{g_{s{1 \atop j,∞}}(x_{1},z_{1})}{g_{s{2 \atop j,∞}}(x_{1},z_{1})}) = \frac{g_{s{1,red \atop j,∞}}(x_{0},z_{0})}{g_{s{2,red \atop j,∞}}(x_{0},z_{0})} (********)

where the red terminology is introduced to indicate that the homogeneous polynomials {g_{s_{j,∞}^{1}}, g_{s_{j,∞}^{2}}} are reduced by any homogeneous factors, vanishing entirely on a component of the projective curve D_{t_{∞},r_{∞}}. By (****), we can pick (x_{0}, z_{0}) to lie on a unique component of D_{t_{∞},r_{∞}}.

Let this component be defined by the homogeneous polynomial h, suppose that g_{s_{j,∞}^{1}} = h^{n}w_{s_{j,∞}^{1}} and g_{s_{j,∞}^{2}} = h^{m}w_{s_{j,∞}^{2}}, for some integers m, n ≥ 0. We can also assume that (x_{0}, z_{0}) does not lie on the intersection D_{t_{∞},r_{∞}} ∩ (w_{s_{j,∞}^{2}} = 0), as this consists of finitely many points on the given component. By the above construction, (x_{1}, z_{1}) does not lie on h = 0, as it does not lie on the curve D_{t_{∞},r_{∞}}. It must, therefore, follow that;

π_{2}(\frac{g_{s{1 \atop j,∞}}(x_{1},z_{1})}{g_{s{2 \atop j,∞}}(x_{1},z_{1})}) = π_{2}[h^{n−m}(x_{1}, z_{1})]π_{2}(\frac{w_{s{1 \atop j,∞}}(x_{1},z_{1})}{w_{s{2 \atop j,∞}}(x_{1},z_{1})}) = \frac{w_{s{1 \atop j,∞}}(x_{0},z_{0})}{w_{s{2 \atop j,∞}}(x_{0},z_{0})}

where, we have used the fact that n = m and, therefore, h^{n−m}(x_{1}, z_{1}) = 1; otherwise, by a simple calculation, using the property (iii)(b), the specialised curve C_{t_{∞}} would contain either one of the points [0 : 0 : 1] or [0 : 1 : 0], contradicting Defn 3.28(i). As (x_{0}, z_{0}) does not lie on any other component of D_{t_{∞},r_{∞}}, we have that;

\frac{w_{s{1 \atop j,∞}}(x_{0},z_{0})}{w_{s{2 \atop j,∞}}(x_{0},z_{0})} = \frac{r(x_{0},z_{0})g_{s{1,red \atop j,∞}}(x_{0},z_{0})}{r(x_{0},z_{0})g_{s{2,red \atop j,∞}}(x_{0},z_{0})} = \frac{g_{s{1,red \atop j,∞}}(x_{0},z_{0})}{g_{s{2,red \atop j,∞}}(x_{0},z_{0})}

where r is the maximal homogeneous factor, of both g_{s_{j,∞}^{1}} and g_{s_{j,∞}^{2}}, vanishing entirely on the remaining components of D_{t_{∞},r_{∞}}. Hence, (********) is shown. Combining (*******) and (********), we then obtain;

π(\frac{g_{s^{1}_{j}}(x_{1},z_{1})}{g_{s^{2}_{j}}(x_{1},z_{1})}) = \frac{g_{s{1,red \atop j,∞}}(x_{0},z_{0})}{g_{s{2,red \atop j,∞}}(x_{0},z_{0})}

Therefore, the limit curve C_{j,t_{∞},r_{∞}} is contained in the union of image curves, defined in (iv)(b). By (****), it consists exactly of the union of these image curves. This show (iv)(a) and (v).

Now, suppose that (x_{0}, y_{0}, z_{0}) belongs to the image curve C_{i,j,∞}, therefore to the limit curve C_{j,t_{∞},r_{∞}}, this means that we can find a generic (t, r, s_{j}, x_{1}, y_{1}, z_{1}) ∈ W_{j}, specialising to (t_{∞}, r_{∞}, s_{j,∞}, x_{0}, y_{0}, z_{0}). As (x_{1}, y_{1}, z_{1}) belongs to C_{j,r,s}, by (iii)(b), the projection pr_{1}(x_{1}, y_{1}, z_{1}) = (x_{1}, y_{1}) belongs to the generic curve C_{t}. We consider the following closed variety Z_{j} ⊂ (S_{j} × P^{2}) defined by;

Z_{j}(t, r, s_{j}, x, y) ≡ C_{t}(x, y)

We have that Z_{j}(t, r, s_{j}, x_{1}, y_{1}), hence, by specialisation, we must have Z_{j}(t_{∞}, r_{∞}, s_{j,∞}, x_{0}, y_{0}), that is, (x_{0}, y_{0}) belongs to the degenerate curve C_{∞}. It follows that the projection, pr_{1} of the image curve C_{i,j,∞} is contained in C_{∞}, (*********).

Conversely, let us suppose that (x_{0}, y_{0}) belongs to the degenerate curve C_{∞}, then, as Z_{j} is irreducible, by a similar argument to the above, we can find a generic (t, r, s_{j}, x_{1}, y_{1}) ∈ Z_{j}, specialising to (t_{∞}, r_{∞}, s_{j,∞}, x_{0}, y_{0}). As (x_{1}, y_{1}) belongs to a generic curve C_{t}, by (iii)(b), we can find a lifting (x_{1}, y_{1}, z_{1}), belonging to C_{j,r,s}. It follows that W_{j}(t, r, s_{j}, x_{1}, y_{1}, z_{1}), and, by specialisation, we must have W_{j}(t_{∞}, r_{∞}, s_{j,∞}, x_{0}, y_{0}, z_{0}), that is (x_{0}, y_{0}, z_{0}) belongs to the limit curve C_{j,t_{∞},r_{∞}}. It follows that C_{∞} is covered by the limit curve C_{j,t_{∞},r_{∞}}, (**********). We can assume that the reduced function η_{j,∞} is non-constant, in which case, we have, by (**********), that the degenerate curve C_{∞} is contained in a line, contradicting the assumption on degree in Definition 3.28. It follows that the projection pr_{1} of an image curve C_{i,j,∞} has dimension 1. As C_{i,j,∞} is irreducible, then, using (*********), we obtain (iv)(b). By similar reasoning to the previous argument , using (**********), we obtain (iv)(d), (^{20}).

Now, suppose that C_{i_{0},j_{0},∞} and C_{i_{1},j_{1},∞} are distinct image curves, we aim to show (iv)(c). Suppose that i_{0} ≠ i_{1}. Then, there exist distinct components {D_{r_{∞}}^{i_{0}}, D_{r_{∞}}^{i_{1}}} of the curve D_{r_{∞}}, and (reduced) birational maps;

η_{j_{0},s_{∞}} : D_{r_{∞}}^{i_{0}} ↭ C_{i_{0},j_{0},∞}

η_{j_{1},s_{∞}} : D_{r_{∞}}^{i_{1}} ↭ C_{i_{1},j_{1},∞} (♯)

If (xyz) ∈ (C_{i_{0},j_{0},∞} ∩ C_{i_{1},j_{1},∞}), then, in particularly, by (♯), (xz) ∈ (D_{r_{∞}}^{i_{0}} ∩ D_{r_{∞}}^{i_{1}}). We, therefore, obtain a map;

pr_{2} : (C_{i_{0},j_{0},∞} ∩ C_{i_{1},j_{1},∞} ∩ A^{3}) → (D_{r_{∞}}^{i_{0}} ∩ D_{r_{∞}}^{i_{1}} ∩ A^{2}) (♯♯)

The fibres of pr_{2} are finite, by the birationality claim in (♯), and the image of pr_{2} is finite, by the distinctness of components claim in (♯).

20(!) The exceptional case when the limit curve D_{t_{∞},r_{∞}} contains the line W = 0 needs to be considered separately. For a general point [1 : λ : 0] on this line, we can find infinitesimals {ϵ_{1}, ϵ_{2}, ϵ_{3}}, with [1+ϵ_{1} : λ+ϵ_{2} : ϵ_{3}], defining a general point on the generic curve D_{t,r}. The image of this point [1 + ϵ_{1} : ϵ_{3}η_{j,s}(\frac{1+ϵ_{1}}{ϵ_{3}} , \frac{λ+ϵ_{2}}{ϵ_{3}} ) : λ + ϵ_{2} : ϵ_{3}], on C_{j,r,s}, projects to [1 + ϵ_{1} : ϵ_{3}η_{j,s}(\frac{1+ϵ_{1}}{ϵ_{3}} , \frac{λ+ϵ_{2}}{ϵ_{3}} ) : ϵ_{3}], on the generic curve C_{t}, and specialises to [1 : 0 : 0] on the limit curve C_{t_{∞}}. We can avoid this degenerate scenario, by imposing the extra conditions specified in footnote 12.

The result (iv)(c), then follows, by observing, (using footnote 20 to exclude the degenerate case), that the intersection of the curve C_{i_{0},j_{0},∞} with the projective plane W = 0, is finite.

Now, suppose that j_{0} ≠ j_{1}, it is sufficient to show that the intersection of the image curves {C_{i_{0},j_{0},∞}, C_{i_{0},j_{1},∞}} is finite. Equivalently, that the reduced functions η_{j_{0},s_{∞}} and η_{j_{1},s_{∞}} define distinct birational morphisms on the component D_{r_{∞}}^{i_{0}}. Suppose that D_{r_{∞}}^{i_{0}} intersects the axis x = 0 at a point (0, c). Let U ⊂ D_{r_{∞}}^{i_{0}} denote an open subset on which the reduced factors {g_{s_{j_{0},∞}}^{1,red}, g_{s_{j_{0},∞}}^{2,red}, g_{s_{j_{1},∞}}^{1,red}, g_{s_{j_{1},∞}}^{2,red}} are nonvanishing. As D_{r_{∞}}^{i_{0}} is irreducible, we can find (x_{0}, z_{0}) ∈ (U ∩ V_{(0,c)}). We can assume that dim(x_{0}, z_{0}/L) = 1, and find an infinitesimal ϵ such that {x_{0}, z_{0}} ⊂ L(ϵ)^{alg}. In particular, there exist infinitesimals {ϵ_{1}, ϵ_{2}} ⊂ L(ϵ)^{alg}, with (x_{0}, z_{0}) = (ϵ_{1}, c + ϵ_{2}). It is sufficient to show that the unique values of the reduced functions {η_{j_{0},s_{∞}}, η_{j_{1},s_{∞}}} are distinct at (x_{0}, z_{0}). We consider the irreducible variety H ⊂ (S_{j_{0},j_{1}} × P^{2}) defined by;

H(t, r, s_{j_{0}}, s_{j_{1}}, x, z) ≡ S_{j_{0},j_{1}}(t, r, s_{j_{1}}) ∧ D_{r}(x, z)

where S_{j_{0},j_{1}} is the irreducible variety, defined by projection of S, similarly to the construction of S_{j}. The intersection H_{ϵ_{1}} ⊂ (S_{j_{0},j_{1}} × P^{2}) is defined by;

H_{ϵ_{1}}(t, r, s_{j_{0}}, s_{j_{1}}, x, z) ≡ H(t, r, s_{j_{0}}, s_{j_{1}}, x, z) ∧ x = ϵ_{1}

We have that H_{ϵ_{1}}(t_{∞}, r_{∞}, s_{j_{0},∞}, s_{j_{1},∞}, x_{0}, z_{0}) and claim that, for (t, r, s_{j_{0}}, s_{j_{1}}) ∈ (S_{j_{0},j_{1}} ∩ V_{(t_{∞},r_{∞},s_{j_{0},∞},s_{j_{1},∞})}), there exists (x_{1}, z_{1}) ∈ P^{2}, with (x_{1}, z_{1}) ∈ V_{(x_{0},z_{0})} and H_{ϵ_{1}}(t, r, s_{j_{0}}, s_{j_{1}}, x_{1}, z_{1}), (♯♯♯). In particular, it follows that H_{ϵ_{1}} consists of a finite union {H_{ϵ_{1}}^{1}, …, H_{ϵ_{1}}^{k}} of irreducible curves, each of which projects onto S_{j_{0},j_{1}}.

The proof of (♯♯♯) is straightforward, using methods from the paper [5]. The tuple (\overline{r}_{∞}, (1, 0, −ϵ_{1})) belongs to the smooth projective space, parametrising intersections between degree n and degree 1 curves. As the intersection (D_{r_{∞}} ∩ (x = ϵ_{1})) is finite, we can apply Theorem 2.4 of [5].

We choose (t, r, s_{j_{0}}, s_{j_{1}}, x_{1}, z_{1}) generically in H_{ϵ_{1}}. In particular, we have that dim(t, r, s_{j_{0}}, s_{j_{1}}/L(ϵ)^{alg}) = 1. It follows that we can find an infinitesimal δ, with {t, r, s_{j_{0}}, s_{j_{1}}} ⊂ L(δ)^{alg} and dim(δ/L(ϵ)^{alg}) = dim(ϵ/L(δ)^{alg}) = 1. We employ the notation above, to produce the commuting specialisation maps;

(π_{1} ∘ π_{2}) : P (L(ϵ, δ)^{alg}) → P (L(ϵ)^{alg}) → P (L).

(π_{2} ∘ π_{1}) : P (L(ϵ, δ)^{alg}) → P (L(δ)^{alg}) → P (L).

We have that π_{2}(x_{1}, z_{1}) = (x_{0}, z_{0}) and π_{1}(x_{0}, z_{0}) = (0, c). Hence, (x_{1}, z_{1}) = (ϵ_{1}, c + ν_{1}), where ν_{1} is an infinitesimal in L(ϵ, δ)^{alg}. As the specialisations commute, we have that (π_{2} ∘ π_{1})(x_{1}, z_{1}) = (π_{1} ∘ π_{2})(x_{1}, z_{1}) = (0, c). It follows that π_{1}(x_{1}, z_{1}) = (0, c + δ_{1}) ∈ D_{t,r}, where δ_{1} is an infinitesimal in L(δ)^{alg}. Let π_{1} define an infinitesimal neighborhood V_{(0,c+δ_{1})}, and let (x_{1}, z_{1}) belong to the branch γ, centred at (0, c + δ_{1}), relative to this neighborhood. Using the condition (iii)(a) and the property of Definition 3.30(iii), by a straightforward adaptation of Lemma 3.27, we have that the values val_{γ}(η_{j_{0},s}) and val_{γ}(η_{j_{1},s}) are distinct, belonging to the fixed set {a_{1}, …, a_{m}} ⊂ L. For convenience of notation, we will denote them by a_{j_{0}} and a_{j_{1}}, (^{21}).

We have that;

η_{j_{0},s}(x_{1}, z_{1}) − η_{j_{1},s}(x_{1}, z_{1}) = (η_{j_{0},s}^{γ}(0, c + δ_{1}) − η_{j_{1},s}^{γ}(0, c + δ_{1}))

+(η_{j_{0},s}(x_{1}, z_{1}) − η_{j_{0},s}^{γ}(0, c + δ_{1}))
−(η_{j_{1},s}(x_{1}, z_{1}) − η_{j_{1},s}^{γ}(0, c + δ_{1}))

By a simple calculation, the terms (η_{j_{0},s}(x_{1}, z_{1}) − η_{j_{0},s}^{γ}(0, c + δ_{1})) and (η_{j_{1},s}(x_{1}, z_{1}) − η_{j_{1},s}^{γ}(0, c + δ_{1})) belong to Ker(π_{1}) ∩ L(ϵ, δ)^{alg}. Hence;

π_{1}(η_{j_{0},s}(x_{1}, z_{1}) − η_{j_{1},s}(x_{1}, z_{1})) = a_{j_{0}} − a_{j_{1}}

and

η_{j_{0},s}(x_{1}, z_{1}) − η_{j_{1},s}(x_{1}, z_{1}) = a_{j_{0}} − a_{j_{1}} + ν

with ν an infinitesimal, belonging to L(ϵ, δ)^{alg}.

21Although it is possible that some permutation of values can occur, for a given function η_{j,s}, at different branches along the axis x = 0.

It follows, using the limit calculation above, that;

π_{2}(η_{j_{0},s}(x_{1}, z_{1}) − η_{j_{1},s}(x_{1}, z_{1})) = η_{j_{0},s_{∞}}(x_{0}, z_{0}) − η_{j_{1},s_{∞}}(x_{0}, z_{0})

= a_{j_{0}} − a_{j_{1}} + ϵ_{3}

where ϵ_{3} is an infinitesimal, belonging to L(ϵ)^{alg}. Therefore;

π_{1}(η_{j_{0},s_{∞}} − η_{j_{1},s_{∞}}) = a_{j_{0}} − a_{j_{1}} ≠ 0

In particular, it follows that the reduced functions η_{j_{0},s_{∞}} and η_{j_{1},s_{∞}} take distinct values at the point (x_{0}, z_{0}). This shows (iv)(c). The proof of (ii) follows immediately from (iv)(b)(d). □

We now show the following fundamental properties of asymptotic degenerations;

Lemma 4.4. Let G be the group obtained in the last section. Then the action of G on a generic curve D_{r}, with r ∈ R, extends uniformly to almost all curves of the form D_{r′} in the family, with r′ ∈ R.

Proof. We can consider G as acting on the curve D_{r}, for generic r ∈ R, replacing C′ in Lemma 3.23. We enumerate the parameters needed to define the action in Lemma 3.23(*), as a tuple;

(g̈_{1}^{1}, g̈_{1}^{2}, …, g̈_{i}^{1}, g̈_{i}^{2}, …, g̈_{l}^{1}, g̈_{l}^{2}) ⊂ P^{2Nl}

where Card(G) = l, which we abbreviate by \overline{g̈} = (g̈_{1}, …, g̈_{i}, …, g̈_{l}), (^{22}). Let

Ψ_{G}(t, r, h̄) ≡ (a).θ_{h_{i}} : D_{r} ↭ D_{r}, 1 ≤ i ≤ l (b).(θ_{h_{i}} ∘ θ_{h_{j}}) = θ_{h_{i∗j}}, 1 ≤ i, j ≤ l

(c).θ_{h_{i}} = Id iff i = 1, 1 ≤ i ≤ l
(d).(θ_{h_{i}})^{−1} = θ_{h_{i^{−1}}}, 1 ≤ i ≤ l

with ∗ and −1 copying the group multiplication and inversion of G onto the labelled set {1, …, l}, and h̄ being a new variable, substituting \overline{g̈}. The conditions (a) and (d) are formulated simultaneously as follows;

22Having obtained a set of 2l rational functions by the standard projective method, we can equalise their degrees, by, for example, multiplying through with factors of the form (x^{s}g(x, z) + 1), where g(x, z) is a polynomial vanishing on the curve D_{r}. This is mainly technically convenient in terms of notation.

∃x_{1} … x_{p}∃z_{1} … z_{p}∀x∀z[⋀_{i=1}^{p}(x ≠ x_{i}) ∧ ⋀_{i=1}^{p}(z ≠ z_{i}) ∧ D_{t,r}(x, z) → ⋀_{i=1}^{w} D_{t,r}(θ_{h_{i}}(x, z))∧⋀_{i=1}^{w}(θ_{h_{i^{−1}}} ∘θ_{h_{i}})(x, z) = (x, z)]

The formulation of conditions (b) and (c)is a similar exercise, which we leave to the reader. By Lemma 3.23, the condition ψ_{G} holds for the tuple (t, r, \overline{g̈}). Imitating the construction above of the variety S, in Lemma 4.3, we can find an irreducible closed variety E ⊂ R × P^{2Nl}, with E → R a finite cover, and open subsets U_{G} ⊂ E and V_{G} ⊂ D, for which the formula Ψ_{G} holds and pr^{−1}(V_{G}) ⊂ U_{G} respectively. If (t′, r′) ∈ V_{G}, then the above construction provides a Galois group action on the curve D_{r′}. By imitating the proof of Lemma 3.23, and using conditions (c) and (d), one can assume that such an action has the generic transitivity property. □

As before, we consider the non-generic case, adopting the terminology from Lemma 4.4;

Lemma 4.5. Let (t_{∞}, r_{∞}, h̄_{∞}) be a non-generic element of E, then, the reduced functions {θ_{h_{1,∞}}^{red}, …, θ_{h_{l,∞}}^{red}} restrict to morphisms from the curve D_{t_{∞},r_{∞}} to itself, satisfying conditions (a) − (d) of the previous lemma, and the generic transitivity property.

Proof. We consider the closed variety J ⊂ (E × P^{2}), defined by;

J(t, r, h̄, x, z) ≡ E(t, r, h̄) ∧ D_{t,r}(x, z)

By similar arguments to the above, J is irreducible. We also consider the variety J_{G} ⊂ (E × P^{2}), defined by;

J_{G}(t, r, h̄, x, z) ≡ V_{G}(t, r) ∧ J(t, r, h̄, x, z) ∧ ∃z_{1} … z_{i} … z_{l}(z = h_{i} ▪ z_{i}) (^{23})

We let \overline{J}_{G} be the Zariski closure of J_{G} in (E × P^{2}). By the construction of V_{G}, a generic fibre \overline{J}_{G}(t, r, h̄) is exactly the curve D_{t,r} ⊂ P^{2}. As V_{G} and D_{t,r} are both irreducible, it follows that J_{G}, and, therefore, \overline{J}_{G} are both irreducible. Hence, J = \overline{J}_{G}, (∗). Now suppose, for contradiction, that the reduced function θ_{h_{1,∞}}^{red} fails to define a morphism from the curve D_{t_{∞},r_{∞}} to itself. Then the intersection D_{t_{∞},r_{∞}} ∩\overline{θ_{h_{1,∞}}^{red}(D_{t_{∞},r_{∞}})} defines a proper closed subset of D_{t_{∞},r_{∞}}. We can choose (x_{0}, z_{0}) from D_{t_{∞},r_{∞}}, not lying on this intersection, (∗∗). By (∗), we have that \overline{J_{G}}(t_{∞}, r_{∞}, h̄_{∞}, x_{0}, z_{0}). As \overline{J_{G}} is irreducible, we can find a generic (t, r, h̄, x_{1}, z_{1}), belonging to \overline{J_{G}}, specialising to (t_{∞}, r_{∞}, h̄_{∞}, x_{0}, z_{0}). By the definition of V_{G}, we can find (x_{1}, z_{2}), belonging to D_{t,r}, with h_{1}▪z_{2} = z_{1}. We can assume that (x_{1}, z_{2}) does not specialise to a point belonging to D_{t_{∞},r_{∞}} ∩ q_{h_{1,∞}}^{red} = 0, (∗ ∗ ∗), where q_{h_{1,∞}}^{red} is the reduced denominator term, used in the definition of θ_{h_{1,∞}}^{red}, see Lemma 3.23. This follows by observing that there are only a finite number of points on the curve D_{t_{∞},r_{∞}}, for which the reduced denominator term is zero. We can choose the coordinate x_{0} so that the line x = x_{0} has an empty intersection with this finite number of points. Therefore, we may assume (∗ ∗ ∗). It follows, by the limit calculation used in Lemma 4.3, that the specialisation (x_{0}, z_{0}) of θ_{h_{1}}(x_{1}, z_{2}) is θ_{h_{1,∞}}^{red}(x_{0}, z_{3}), with (x_{0}, z_{3}) ∈ D_{t_{∞,r∞}}. This contradicts (∗∗). Hence, the first part of the lemma is shown. We now check conditions (a) − (d). Condition (a) follows from condition (d) and the fact that the set of reduced functions {θ_{h_{1,∞}}^{red}, …, θ_{h_{l,∞}}^{red}} define morphisms on the curve D_{t_{∞},r_{∞}}. Conditions (b) and (d) amount to checking that for a triple (t, r, ḡ), specialising to (t_{∞}, r_{∞}, ḡ_{∞}), for a generic point (x_{1}, z_{1}) of D_{t,r}, specialising to (x_{0}, z_{0}) of D_{t_{∞},r_{∞}}, we have that;

23The reader should supply the extra conditions necessary to exclude the finite sets of points for which the action ▪ of G is undefined, a similar argument being used in Lemma 4.3

sp(θ_{h_{i}} ∘ θ_{h_{j}}(x_{1}, z_{1})) = θ_{h_{i,∞}}^{red} ∘ θ_{h_{j,∞}}^{red}(x_{0}, z_{0}) (∗ ∗ ∗∗)

We assume that the three points {(x_{1}, z_{1}), θ_{h_{j}}(x_{1}, z_{1}), θ_{h_{i}} ∘θ_{h_{j}}(x_{1}, z_{1})} satisfy (∗ ∗ ∗), by excluding a finite number of the coordinates x_{0}. In which case, we have, by repeating the limit calculation in Lemma 4.3, that;

sp(θ_{h_{i}} ∘ θ_{h_{j}}(x_{1}, z_{1})) = θ_{h_{i,∞}}^{red}(sp(θ_{h_{j}}(x_{1}, z_{1})))

= θ_{h_{i,∞}}^{red}(θ_{h_{j,∞}}^{red}(sp(x_{1}, z_{1})))

= θ_{h_{i,∞}}^{red} ∘ θ_{h_{j,∞}}^{red}(x_{0}, z_{0})

as required in (∗ ∗ ∗∗). One direction of condition (c) is straightforward and left to the reader. We show the converse direction, (∗ ∗ ∗ ∗ ∗). We let E_{i}, for 1 ≤ i ≤ l, be the projections of E onto the l factors of G. As before, the covers E_{i} → R are irreducible and finite. We also introduce the combined parameter spaces; S_{j,k}(t, r, s_{j}, s_{k}) ≡ S_{j}(t, r, s_{j}) ∧ S_{k}(t, r, s_{k}), for 1 ≤ j, k ≤ m

F_{i,j,k}(t, r, h_{i}, s_{j}, s_{k}) ≡ E_{i}(t, r, h_{i}) ∧ S_{j,k}(t, r, s_{j}, s_{k}), for 1 ≤ i ≤ l, 1 ≤ j, k ≤ m

Again, the covers S_{j,k} → R and F_{i,j,k} → R are irreducible and finite. We recall, from Lemma 3.23, that, for a generic parameter (t, r) ∈ R, any g ∈ G, acting on the curve D_{t,r}, induces an action on the associated set of flashes {C_{1,r,s}, …, C_{j,r,s}, … C_{m,r,s}}. If θ_{g} ≠ Id, then, by definition of the Galois group, this action is non-trivial. By the enumeration in the previous lemma, we denote g by g_{2} ≠ g_{1}, and ,without loss of generality, we can suppose that (g_{2} ▪ C_{1,r,s}) = C_{2,r,s}. We recall the definition of the variety T_{1} ⊂ F_{2,1,2} × P^{3}, considered in Lemma 4.3;

T_{1}(t, r, h_{2}, s_{1}, s_{2}, x, y, z) ≡ F_{2,1,2}(t, r, h_{2}, s_{1}, s_{2})∧D_{t,r}(x, z)∧y = η_{1,s_{1}}(x, z)

and define the following variety T_{2,g_{2}} ⊂ F_{2,1,2} × P^{3};

T_{2,g_{2}}(t, r, h_{2}, s_{1}, s_{2}, x, y, z) ≡ F_{2,1,2}(t, r, h_{2}, s_{1}, s_{2}) ∧ D_{t,r}(x, z)

∧ y = (η_{2,s_{2}} ∘ θ_{2,h_{2}})(x, z) (^{24})

As we showed in Lemma 4.3, the Zariski closure \overline{T_{1}} is irreducible. By the condition that (θ_{2,h_{2}})^{∗}(η_{2,s_{2}}) = η_{1,s_{1}} on the curve D_{t,r}, for a generic choice of (t, r, h_{2}, s_{1}, s_{2}), we clearly have that T_{1} = T_{2,g_{2}}, and the closure \overline{T_{2,g_{2}}} is irreducible. By results in Lemma 4.3, a degenerate fibre \overline{T_{1}}(t_{∞}, r_{∞}, h_{2,∞}, s_{1,∞}, s_{2,∞}) is exactly the limit curve, C_{1,t_{∞,r∞}}. Suppose (x_{0}, y_{0}, z_{0}) ∈ C_{1,t_{∞,r∞}}, then we can choose a generic point (t, r, h_{2}, s_{1}, s_{2}, x_{1}, y_{1}, z_{1}) on the variety \overline{T_{2,g_{2}}}, specialising to the tuple (t_{∞}, r_{∞}, h_{2,∞}, s_{1,∞}, s_{2,∞}, x_{0}, y_{0}, z_{0}). By the definition of T_{2,g_{2}}, we have that y_{1} = (η_{2,s_{2}} ∘ θ_{2,h_{2}})(x_{1}, z_{1}). Repeating the above calculation;

y_{0} = sp(y_{1}) = sp((η_{2,s_{2}} ∘ θ_{2,h_{2}})(x_{1}, z_{1}))

= η_{2,s_{2,∞}}^{red}(sp(θ_{2,h_{2}}(x_{1}, z_{1})))

= η_{2,s_{2,∞}}^{red}(θ_{2,h_{2,∞}}^{red}(sp(x_{1}, z_{1})))
= (η_{2,s_{2,∞}}^{red} ∘ θ_{2,h_{2,∞}}^{red})(x_{0}, z_{0})

24It is necessary to supply extra conditions, in order to ensure both varieties are well defined. This was done in Lemma 4.3, for the variety T_{1}. For the variety T_{2,g_{2}}, one should restrict the parameter x, by adding the conditions that x ∉ Disc_{x′}(D_{t,r}(x′, z), q_{h_{2}^{2}}(x′, z) = 0) and x ∉ Disc_{x′}(D_{t,r}(x′, z), g_{s_{2}^{2}}(x′, z) = 0)

where, we take the usual precaution, of restricting the range of the coordinate x_{0}, in order to apply the calculation of Lemma 4.3. The result implies that, in the limit, (θ_{2,h_{2,∞}}^{red})^{∗}(η_{2,s_{2,∞}}) = η_{1,s_{1,∞}} on the curve D_{t_{∞},r_{∞}}. By Lemma 4.3(c), we have that the rational functions {η_{1,s_{1,∞}}, η_{2,s_{2,∞}}} are distinct. Hence, we conclude, that θ_{2,h_{2,∞}}^{red} ≠ Id. This shows (∗ ∗ ∗ ∗ ∗).

We finally show the generic transitivity property, for the curve D_{t_{∞},r_{∞}}. Consider the variety F ⊂ R × P^{1} × P^{2};

F (t, r, x_{0}, x_{1}, X, Z, W) ≡ D_{t,r}(X, Z, W) ∧ x_{0}X = x_{1}W

which defines a finite cover F → (R × P^{1}). Let Q_{1} = [0 : 1 : 0], and let k = mult_{(t_{∞},r_{∞},a,Q_{1})}(F/(R × P^{1}), where a is generic over the parameter (t_{∞}, r_{∞}), and the parameters defining R. For generic (t, r, a) ∈ (R × P^{1}), we have that I_{Q_{1}}(D_{t,r}, x = a) = n − l, where Card(G) = l and deg(D_{t,r}) = n. This follows, from a simple application of Bezout’s theorem, and the previously observed fact, that the curve D_{t,r} has l points in finite position, along the axis x = a. It follows that I_{Q_{1}}(D_{t_{∞},r_{∞}}, x = a) = n − l + (k − 1) = n − (l − k + 1). This follows from summability of specialisation, see [5]. Applying Bezout’s theorem again, the curve D_{t_{∞},r_{∞}} has exactly (l − k + 1) points in finite position along the generic axis x = a. If k > 1, we obtain l_{1} < l points in finite position along a generic axis. By the counting argument, used in Lemma 3.23, properties (a) − (d) shown above, and the fact that Card(G) = l, we can find a reduced function θ_{h_{i,∞}}^{red}, for some (i > 2), acting as the identity on D_{t_{∞},r_{∞}}, this contradicts property (c). Hence, k = 1, and the curve D_{t_{∞},r_{∞}} has exactly l points in finite position along a generic axis. Transitivity of the G-action, then follows by applying the methods of Lemma 3.23, (^{25}) □

We now observe the following uniformity in results from the previous section.

Lemma 4.6. Let g(x, z) be chosen, satisfying the conditions of Lemma 3.4, with corresponding θ_{g}, then θ_{g} defines a birational morphism on each curve in the family D_{r}, for r ∈ R. Moreover, we can find a new space of parameters R′, satisfying conditions 4.3(i),(ii), S′ satisfying conditions 4.3(iii),(iv),(v), such that the algebraic family D_{r′}, for r′ ∈ R, consists of the images of θ_{g} : D_{r} ↭ D_{r′}, the functions η_{j,r′,s′} = (θ_{g}^{−1})^{∗}(η_{j,r,s}), for generic (r, s), η_{j,r′_{∞},s′_{∞}}^{red} = (θ_{g}^{−1})^{∗}(η_{j,r_{∞},s_{∞}}^{red}), for degenerate (r_{∞}, s_{∞}) and E′, satisfying Lemma 4.4, with θ_{r′,h̄′} = (θ_{g}^{−1})^{∗}(θ_{r,h̄}), for generic (r, h̄), and θ_{r′_{∞},h̄′_{∞}}^{red} = (θ_{g}^{−1})^{∗}(θ_{r_{∞},h̄_{∞}}^{red}), for degenerate (r_{∞}, h̄_{∞}). There exists a naturally defined total morphism from the unprimed to the primed parameter spaces, Γ_{g} : (R, S, E) → (R′, S′, E′), such that all relevant diagrams commute.

25For a general 1-dimensional family of curves of fixed projective degree, all of which are irreducible, it is not necessarily true, that deg(C_{t_{0}}) = deg(C_{t_{1}}), for a pair (t_{1}, t_{2}), where deg(C) denotes deg(L(C)/L(x)). It is an interesting question to determine the conditions for which this property holds. The family parametrised by R is close to having this property, depending on whether the degenerate fibres can be chosen as irreducible.

Proof. The proof is straightforward but lengthy. That θ_{g} defines a birational morphism on each curve in the family D_{r}, for r ∈ R, follows easily from the proof of Lemma 3.4, (^{26}). In order to construct the parameter space R′, observe that we can write the equation of a general member of the family D_{r}, in the form;

∑_{0≤i+j≤n} r_{ij}x^{i}z^{j} = 0 (∗)

Now make the substitution z = g(x, z′) in (∗) and clear the denominator term, to obtain the expression;

∑_{0≤i+j≤n} r_{ij}x^{i}[a(x)z′ + b(x)]^{j}[c(x)z′ + d(x)]^{n−j} = 0 (∗∗)

We can then expand (∗∗), obtaining an expression of the form;

∑_{0≤i+j≤n′} r′_{ij}x^{i}z^{′j} (∗ ∗ ∗)

in which each r′_{ij} term is a linear combination of terms r_{ij}, with coefficients depending on the those of the polynomials {a(x), b(x), c(x), d(x)}. Comparing (∗) and (∗ ∗ ∗), we obtain a linear projective morphism Γ′ from R ⊂ P^{N} to R′ ⊂ P^{N′}, which parametrises an irreducible 1dimensional family of curves, with degree n′. By construction, we have that θ_{g} : D_{r} ↭ D_{r′}. The family R′ satisfies condition 4.3(i), by the proof of Lemma 3.4. The construction of S′ is similar, by substitution of the polynomial g^{−1}(x, z′), defining the inverse transformation θ_{g}^{−1}, for z, in the rational functions {g_{s_{j}^{1}}(x, z), g_{s_{j}^{2}}(x, z)}. The statement concerning the transformation of η_{j,r,s}, for generic (r, s) follows immediately from this method, while the statement concerning the transformation of η_{j_{∞},r_{∞},s_{∞}}^{red}, for degenerate (r_{∞}, s_{∞}), is a limit calculation, of a form we have considered above. We leave the construction of E′ and the associated transformations of the Galois action to the reader. It is clear how the total morphism Γ_{g} is obtained from the partial linear projective morphisms on the spaces {R, S, E}. □

26One can easily exclude the exceptional case that θ_{g} is not uniformly and birationally defined, namely that a degenerate curve D_{r_{∞}} has a component l_{∞}, by imposing the extra conditions in footnote 12

Remarks 4.7. The above construction can be repeated for any abstract birational map θ_{g}, which is uniformly birational on the curves in the family D_{r}, for r ∈ R, and preserves the conditions of Lemma 3.3, for a generic curve, see also footnote 3. When referring to birational maps on a degenerate fibre, we will assume that these extra conditions are satisfied. In particular, we always have that (pr ∘ θ_{g}) = pr uniformly.

As a consequence of the lemma, we are able to extend Definition 3.11, to the case of a degenerate curve D_{t_{∞},r_{∞}};

Definition 4.8. We define a branch γ of D_{t_{∞},r_{∞}} to be blue, if there exists a birational map θ : D_{t_{∞},r_{∞}} ↭ D_{t_{∞},r′_{∞}}, see previous remark, such that the either/or clause of Definition 3.11 holds, for D_{t_{∞},r′_{∞}} and θ(γ). We define a branch γ to be silver otherwise.

Remarks 4.9. This is a good definition, using the same argument as in Remarks 3.12(b). We can also give an equivalent definition of a silver branch, using Remarks 3.12(c).

We can now use uniformity of the G-action, to simplify the geometry of the degenerate curve.

Lemma 4.10. Lemma 3.13 holds for D_{t_{∞},r_{∞}}, that is the distribution of silver and blue branches along any axis is uniform.

Proof. The result follows immediately from the proof of Lemma 3.13 and uniformity of the G-action, Lemma 4.5.

□

and to exclude symmetry properties;

Lemma 4.11. Lemma 3.34 holds for D_{t_{∞},r_{∞}}. That is, for a blue branch γ in finite position, it is not the case that every function {η_{1,t_{∞},r_{∞}}^{red}, …, η_{m,t_{∞},r_{∞}}^{red}} is symmetric at γ.

Proof. Using Lemma 3.34, Lemma 4.3(iv)(c) and Lemma 4.5.

□

5. Asymptotic Degenerations to Lines Passing through a Single Point

We consider a special case of the families considered in the previous section, for which a degenerate fibre C_{t_{∞}} of {C_{t}}_{t∈P^{1}}, consists of m lines, passing through a single point O. It is convenient to assign the point O with the coordinates (k, 0), for k ≠ 0, in the (x, y) plane. The m distinct lines must then correspond to l_{[(0,a_{j}),(k,0)]}, for 1 ≤ j ≤ m, with equations (a_{j} − 1)x + ky = 0. The analysis of degenerate flash intersections is easier due to the simplified geometry of this configuration. In particular, the image curves C_{i,j,∞} of Lemma 4.3, are all contained in the union of planes formed by the m distinct lines and the point Q_{1} ∈ P^{3}. We make the following definition;

Definition 5.1. Let ℓ = {l̆_{1}, …, l̆_{m}} enumerate the branches of C_{t_{∞}}, which are centred at O.

Definition 5.2. For the branches in Definition 5.1, we define Υ_{j} = {γ̆_{1,j}, …, γ̆_{s,j}} to be the lifting of these branches to the flash C_{j,∞} = ⋃_{1≤i≤t} C_{i,j,∞}, and also to the curve D_{t_{∞},r_{∞}}. We define Υ = ⋃_{1≤j≤m} Υ_{j}.

Remarks 5.3. We emphasise the point that, for any j, every branch in ℓ lifts to a branch in Υ_{j}, this follows immediately from Lemma 4.3(iv)(d)

We have the same birational invariance of such branches, as in Lemma 3.27;

Lemma 5.4. Let θ : D_{t_{∞},r_{∞}} ↭ D_{t_{∞},r′_{∞}} be a birational map, in the sense of Remarks 4.7, then the branches {γ̆, η_{j,s_{∞},t_{∞}}^{red}(γ̆)} of {D_{t_{∞},r_{∞}}, C_{j,s_{∞}}} lift the same branch l̆on C_{t_{∞}} as {θ(γ̆), (η_{j,s′_{∞},t_{∞}}^{red}∘θ∘pr_{2})(γ̆)} of D_{t_{∞},r′_{∞}}, C_{j,s′_{∞}}. In particular, the sets Υ_{j} and Υ are birationally invariant.

Proof. The proof is the same as Lemma 3.27. □

We have a stronger version of Lemmas 3.30 and 3.31;

Lemma 5.5. Let γ̆ be a geometric branch of D_{t_{∞},r_{∞}}. Then;

valγ̆(η_{j_{0},s_{∞},t_{∞}}^{red}) = valγ̆(η_{j_{1},s_{∞},t_{∞}}^{red}), for j_{0} ≠ j_{1}, iff γ̆ ∈ (Υ_{j_{0}} ∩ Υ_{j_{1}}) iff γ̆ ∈ ⋂_{1≤j≤m} Υ_{j}

Moreover, Υ = Υ_{j}, for 1 ≤ j ≤ m.

Proof. The first two equivalences follow by using the proof of Lemma 3.30, together with the fact that D_{t_{∞},r_{∞}} has a unique singularity at O. In order to show the third equivalence, suppose that the branch γ̆ is in finite position along the axis x = k, with valγ̆(η_{j_{0},s_{∞},r_{∞}}^{red}) = valγ̆(η_{j_{1},s_{∞},r_{∞}}^{red}) = 0. Suppose that valγ̆(η_{j_{2},s_{∞},r_{∞}}^{red}) = a ≠ 0, for some j_{2} ∉ {j_{0}, j_{1}}. We can assume that this value is finite, in which case we obtain that (k, a) belongs to the curve D_{t_{∞},r_{∞}}. This is clearly impossible. Hence, valγ̆(η_{j_{2},s_{∞},r_{∞}}^{red}) = 0, and, γ̆ belongs to Υ_{j_{2}}. For the final statement, simply observe that if, without loss of generality, γ̆ belongs to Υ_{1}, then γ̆ is centred along the axis x = k, and valγ̆(η_{1,s_{∞},r_{∞}}^{red}) = 0. If valγ̆(η_{j,s_{∞},r_{∞}}^{red}) = a ≠ 0, for j ≠ 1, we again obtain a point (k, a) belonging to the curve D_{t_{∞},r_{∞}}, which is a contradiction. Therefore γ̆ ∈ ⋂_{1≤j≤m} Υ_{j}, and Υ = Υ_{j}, for 1 ≤ j ≤ m.

□

We obtain the following characterisation of branches for the degenerate curve;

Lemma 5.6. All the branches of D_{t_{∞},r_{∞}} are silver.

Proof. Using the proofs of Lemma 3.36 and Lemma 3.37. □

We now show that we can simultaneously isolate silver branches, and refer to the previous section for notation;

Lemma 5.7. Suppose that γ is a silver branch, in finite position, of the generic curve D_{t,r}, centred at (k′, a′), specialising to the point (k, a) of the degenerate curve D_{t_{∞},r_{∞}}. Then, after a sequence of birational transformations, we can assume that γ is isolated and there exists a unique silver branch of D_{t_{∞},r_{∞}}, centred at (k, a).

Proof. We follow the proof of Lemma 3.14. Let (k′ + x, a′ + η_{1}(x)) be a parametrisation of γ, where η(x) is a power series belonging to K[[x]], with sp(P (K)) = P (L), of the form;

η(x) = ∑_{j=1}^{∞} a′_{j}x^{j}

Suppose that the sequence (a′, a′_{1}, …, a′_{j}, …) specialises to the sequence (a, a_{1}, …, a_{j}, …). We define;

g(x, z) = \frac{(z−a)}{(x−k)} − \frac{a}{k}

and apply the birational map θ_{g}, satisfying Lemma 4.6. Let {γ̆_{1}, …, γ̆_{r}} enumerate the branches of D_{t_{∞},r_{∞}}, centred at (k, a), then, using Lemma 5.6, the only remaining branches, in finite position along the axis x = k, after applying this transformation, are {θ_{g}(γ̆_{1}), …, θ_{g}(γ̆_{r})}. Their centres depend only on the first coefficient b_{1} of the power series {η_{1}(x), …, η_{r}(x)} defining these branches. We consider the effect of θ_{g} on γ. The power series representation of θ_{g}(γ) is given by;

(k′ + x, \frac{(a′−a)+η(x)}{(k′−k)+x} − \frac{a}{k}) = (k′ + x, \frac{ϵ_{1}+η(x)}{ϵ_{2}+x} − \frac{a}{k})

so, the centre of γ_{1} = θ_{g}(γ), specialises to (k, a_{1} − \frac{a}{k}). By elementary properties of specialisations, one of the branches {γ̆_{1}, …, γ̆_{r}} must therefore be centred at (k, a_{1} − \frac{a}{k}) = (k, c_{1}), requiring that b_{1} = a_{1}. We repeat the process for the new function;

g_{1}(x, z) = \frac{(z−c_{1})}{(x−k)} − \frac{c_{1}}{k}

Now the centre of θ_{g_{1}}(γ_{1}), specialises to (k, a_{2} − \frac{c_{1}}{k}). Again, one of the remaining branches in \{γ̆_{1}, …, γ̆_{r}\} must be centred at (k, a_{2}− \frac{c_{1}}{k}) = (k, c_{2}), requiring that, the second coefficient of the defining series b_{2} = a_{2}. We repeat the process, until we are left with a single silver branch γ̆_{1} say, along the axis x = k, for the curve D_{t_{∞},r_{∞}}. The original branch γ is also isolated, using, for example, preservation of singularities, see Remarks 5.11. Hence, the lemma is shown

□

Remarks 5.8. As an immediate consequence of the proof, we see that for the remaining silver branch γ̆, the original power series representation is given by;

a + η̆(x), where η̆(x) = ∑_{j=1}^{∞} a_{j}x^{j}

By applying the finite number of birational transformations, in the previous lemma, to the series (a′ + η(x)) and (a + η̆(x)), we obtain, in the final situation, that the defining series of {γ, γ̆} are related by specialisation.

We make the following definition;

Definition 5.9. By a general nodal degeneration, we mean a family of curves {C_{t} : t ∈ P^{1}}, satisfying (i) − (iv) of the conditions at the beginning of Section 1, together with (v′). C_{∞} is a union of m distinct lines {l_{1}, …, l_{m}}.

A desirable property of general nodal degenerations is the following;

Definition 5.10. We say that a general nodal degeneration has good specialisation, if, for any given node of the generic curve C_{t}, centred at (a′, b′), and specialising to a point (a, b), formed by the intersection of two lines {l_{1}, l_{2}} of the degenerate curve C_{∞}; if {c′, d′} denote the gradients of the tangent lines to the two branches of the node on C_{t}, then the specialisations {c, d} are the gradients of the two lines {l_{1}, l_{2}}.

Remarks 5.11. It is always true that the point (a′, b′) specialises to a point (a, b) formed by the intersection of two lines. This is simply a consequence of the preservation of singularities. More specifcally, if the family of curves is determined by a polynomial;

F (X, Y, W; t) = ∑_{i+j+k=m} a_{ijk}(t)X^{i}Y^{j}Z^{k}

Then the nodes of C_{t} are determined by the intersections of the varieties \frac{∂F}{∂X} = \frac{∂F}{∂Y} = \frac{∂F}{∂Z} = 0. For a point (a′, b′) belonging to this intersection, the specialisation also belongs to this intersection at t = ∞. This is clearly a singular point of the variety C_{∞}, hence, a point formed by the intersection of two lines. However, the stronger property concerning specialisation of gradients is not necessarily true; Zariski gives a counterexample in his paper [28], refuting Severi’s original proof of his conjecture.

We show how to correct this error;

Theorem 5.12. An asymptotic general nodal degeneration has good specialisation.

Proof. We will consider only the special case when the degenerate fibre consists of m distinct lines passing through a point O. The more general case of m distinct lines is left as an exercise for the reader. Let {ρ_{1}, ρ_{2}} be branches of a given node of the generic curve C_{t}. By Lemmas 3.36 and 3.43, we can find a silver branch γ ∈ D_{t,r}, which is isolated in finite position, and such that, without loss of generality, η_{1,r,s}(γ) and η_{2,r,s}(γ) lift the branches ρ_{1} and ρ_{2} respectively, with value ϵ. We have, using, for example, Remarks 5.11, that γ is centred at a point (k′, a′), specialising to (k, a). By Lemma 5.7, we can assume that there exists a unique silver branch γ̆ of the degenerate curve D_{t_{∞},r_{∞}}, centred at (k, a). The point (t, r, s, k′, ϵ, a′) belongs to the closed variety W_{1,2}, where W_{j} is defined in Lemma 4.3;

74 TRISTRAM DE PIRO

W_{1,2}(t, r, s, x, y, z) ≡ W_{1}(t, r, s, x, y, z) ∧ W_{2}(t, r, s, x, y, z) Hence, so does the specialisation (t_{∞}, r_{∞}, s_{∞}, k, 0, a). It follows from

the description in Lemma 4.3(iv), that; valγ̆(η_{1,r_{∞},s_{∞}}^{red}) = valγ̆(η_{2,r_{∞},s_{∞}}^{red}) = 0 By Lemma 5.5, it follows that η_{1,r_{∞},s_{∞}}^{red}(γ̆) and η_{2,r_{∞},s_{∞}}^{red}(γ̆) lift the

branches l̆_{1} and l̆_{2} respectively, where {l̆_{1}, l̆_{2}} are the branches formed by the intersection of a pair of lines at the origin O.

We now compute the slopes of the branches ρ_{1} and ρ_{2}, denoted {c′, d′}. Recall that the plane curves D_{r}(x, z) are parametrised by;

F (x, z; r) = ∑_{i+j≤k} r_{ij}x^{i}z^{j} and the rational function η_{1,r,s}(x, z) = \frac{g_{s^{1}_{1}}(x,z)}{g_{s^{2}_{1}}(x,z)}. We have that; c′ = D(η_{1,r,s})_{(k′,a′)} ▪ D(F_{r})_{(k′,a′)} = \frac{∂η_{1,r,s}}{∂x} \frac{∂F_{r}}{∂x} |_{(k′,a′)} + \frac{∂η_{1,r,s}}{∂y} \frac{∂F_{r}}{∂y} |_{(k′,a′)} d′ = D(η_{2,r,s})_{(k′,a′)} ▪ D(F_{r})_{(k′,a′)} = \frac{∂η_{2,r,s}}{∂x} \frac{∂F_{r}}{∂x} |_{(k′,a′)} + \frac{∂η_{2,r,s}}{∂y} \frac{∂F_{r}}{∂y} |_{(k′,a′)} Using a similar limit calculation to the above, (^{27}), we have; sp(c′) = sp(D(η_{1,r,s})_{(k′,a′)} ▪D(F_{r})_{(k′,a′)}) = D(η_{1,r_{∞},s_{∞}}^{red})_{(k,a)} ▪D(F_{r_{∞}})_{(k,a)} sp(d′) = sp(D(η_{2,r,s})_{(k′,a′)}▪D(F_{r})_{(k′,a′)}) = D(η_{2,r_{∞},s_{∞}}^{red})_{(k,a)}▪D(F_{r_{∞}})_{(k,a)} By the same reasoning as above, we see that sp(c′) = c and sp(d′) = d

as required. □

It is clearly impossible to obtain an asymptototic degeneration, in the sense of Definition 4.1, to a set of lines passing through a single point O, without imposing the additional assumption on C, that all of the tangent lines to the branches along the axis x = 0 pass through the point O. However, we can find a weaker version of Definition 4.1(iii), for which the results we have shown still hold, namely;

27The differential calculation should be checked carefully, one should compute the differential of \frac{g_{s^{1}_{1}} (x,z)}{g_{s^{2}_{1}} (x,z)} by parts, first, and then, as was done above, find the specialisation of this function, without specialising the corresponding point. It is easily checked that this specialisation is the same as the differential of η_{1,r_{∞},s_{∞}}^{red}

(iii)’. Almost every curve C_{t}, has m distinct non-singular branches, centred at the points {(0, a_{1}), …, (0, a_{m})}. We will also refer to this condition as asymptotic.

The specific construction of asymptotic degenerations to lines through a given point, follows using a projective method, generally referred to as Severi’s cone construction. We consider a class of degenerations, obtained by projecting a fixed irreducible curve C ⊂ P^{3}, onto a plane H ⊂ P^{3}, from a variable point Q. We first show the following lemma;

Lemma 5.13. Let C ⊂ P^{3} be an irreducible projective curve, not contained in a hyperplane section. Let ω, l be a hyperplane and a line, respectively, with l ⊊ ω, and (l ∩ ω) = Q. Let P ar_{m} = P^{N} denote the parameter space for plane projective curves of degree m, where \frac{N=m(m+3)}{2}. Then, there exist morphisms {θ, ϕ};

θ : P^{1} → P ar_{t}, C_{t} defined by θ(t)

ϕ : P^{1} ≅ l, ϕ(t) = Q_{t}, Q = Q_{0}

with the property that, for t ≠ 0, C_{t} is the plane projective curve, obtained as the projection of C from Q_{t} to ω, see the remarks at the beginning of Section 4 of [6].

Proof. Without loss of generality, by making a linear change of variables, we can suppose that the coordinates of P^{3} are given by {X, Y, Z, W } and {X, Y, W } denote the coordinates of the hyperplane ω, embedded into P^{3} by the map;

i : P^{2} → P^{3}

[X : Y : W] ↦ [X : Y : 0 : W]

We denote the restriction of these coordinates to A^{2} and A^{3} by (x, y) and (x, y, z) respectively, and we have a corresponding inclusion;

i : A^{2} → A^{3}

(x, y) ↦ (x, y, 0)

We can assume that Q corresponds to the origin (0, 0, 0) of the coordinate system (x, y, z), with projective coordinates [0 : 0 : 0 : 1], and l corresponds to the line x = y = 0. Let C be defined by the affine equations F_{1}(x, y, z) = F_{2}(x, y, z) = 0, and let (0, 0, t) parametrise a point Q_{t}, along the line l. Let;

C_{t}(x, y) ≡ ∃λ∃z_{1}∃z_{2}∃z_{3}[F_{1}(z_{1}, z_{2}, z_{3}) = F_{2}(z_{1}, z_{2}, z_{3}) = 0 ∧λ(z_{1}, z_{2}, z_{3})+(1−λ)(0, 0, t) = (x, y, 0)] (∗)

By the assumptions on C, the construction in Section 4 of [6], and the definition (∗), for t ≠ 0, \overline{C_{t}(x, y)} = pr_{Q_{t}}(C), where the closure is taken inside the projective plane ω or P^{2}. We let \overline{C} ⊂ A^{3} denote the closure of C(x, y, t). By elementary dimension considerations, and the fact that the generic fibre C_{t}(x, y) is irreducible, being an open subset of the irreducible curve pr_{Q_{t}}(C), \overline{C} is also irreducible. One can, therefore, find a single equation G(x, y, t) = 0, defining the surface \overline{C}. By making the substitutions x = \frac{X}{W} and y = \frac{Y}{W}, we can find a homogeneous polynomial H(X, Y, W, t), of degree m, in the variables {X, Y, W }, whose restriction to A^{3}, corresponds to G(x, y, t). We define θ, as a morphism to P ar_{m}, by taking the coefficients of the polynomial H. It is a simple exercise to check that the conditions of the lemma are satisfied.

□

We observe the following further property of the degenerations, given in Lemma 5.13.

Lemma 5.14. Projective Degenerations

Let C ⊂ P^{3} be an irreducible projective curve and ω a hyperplane, intersecting C transversely in m distinct points {P_{1}, …, P_{m}}, where m = deg(C). Let l be a line, intersecting ω in a new point Q, then the curve C_{0} consists of the union of lines ⋃_{1≤j≤m} l_{P_{j}Q}.

Proof. We show that the curves C_{t} are obtained as the intersection of ω, with the cone Cone_{Q_{t}}(C), for t ∈ P^{1}, (∗). We, first, consider the case, that t ≠ 0, and, without loss of generality, assume that Q_{t} ∉ C, the case when Q_{t} ∈ C being left to the reader. It is a straightforward exercise, to check that;

Cone_{Q_{t}}(C) = ⋃_{w∈C} l_{Q_{t}w}

is a closed, irreducible,subvariety of P^{3}. The fact that pr_{Q_{t}}(C) = (Cone_{Q_{t}}(C) ∩ ω) follows immediately from the definitions of the projection pr_{Q_{t}}, and the cone Cone_{Q_{t}}(C). As before, we assume that C is defined by the affine equations F_{1}(x, y, z) = F_{2}(x, y, z) = 0, Q_{t} is parametrised by (0, 0, t) and ω is defined by z = 0, in the coordinate system (x, y, z). Let;

D_{t}(x, y, z) ≡ ∃λ∃z_{1}∃z_{2}∃z_{3}[F_{1}(z_{1}, z_{2}, z_{3}) = F_{2}(z_{1}, z_{2}, z_{3}) = 0 ∧ λ(z_{1}, z_{2}, z_{3}) + (1 − λ)(0, 0, t) = (x, y, z)]

Let \overline{D} ⊂ A^{4} denote the closure of D(x, y, z, t). As before, using the fact that the generic fibre D_{t}(x, y, z) is an open subset of the irreducible variety Cone_{Q_{t}}(C), \overline{D} is irreducible and we can find a single equation G(x, y, z, t) = 0, defining the surface \overline{D}. By projectivizing the polynomial G, we can find a single homogeneous polynomial G(X, Y, Z, W, t), parametrising the cones Cone_{Q_{t}}(C). We let;

H(X, Y, W, t) ≡ G(X, Y, 0, W, t)

By definition, H parametrises the intersections (Cone_{Q_{t}}(C)∩ω). It is a simple exercise, to check that this is a different definition of the same variety, given in Lemma 8.1, hence (∗) is shown. In order to complete the proof, it is sufficient to compute the intersection (Cone_{Q_{0}}(C)∩ω). If w ∈ (C \ω), then (l_{wQ_{0}} ∩ω) = Q_{0}, whereas, if w ∈ (C ∩ω), then (l_{wQ_{0}} ∩ ω) = l_{wQ_{0}}, using the fact that Q_{0} ∈ ω. The result follows immediately, from the fact that (C ∩ω) = {P_{1}, …, P_{m}}, and distinctness of the point Q = Q_{0}. □

The above lemmas allow us to easily construct degenerations to lines through a given point. In order to obtain the extra asymptotic conditions, for curves of degree 2, we require the following variant;

Lemma 5.15. Mirror Construction

Let \{ω_{1}, ω_{2}\} ⊂ P^{3} be two planes intersecting in a line l. Let \overline{C} ⊂ ω_{1}, with the property that pr_{P} (\overline{C}) = C, relative to the hyperplane ω_{2} ⊂ P^{3}, and l intersects C transversely at the points {P_{1}, P_{2}}, where P is generic in P^{3}. Let Q = l_{P_{1}} ∩l_{P_{2}}, where l_{P_{1}} and l_{P_{2}} are the tangent lines of C to {P_{1}, P_{2}} respectively. Let l′ be a line, passing through P, and intersecting ω_{1} at Q. Then the set of projections; pr_{x}(\overline{C}) : x ∈ (l′ \ Q)\}

together with the limit curve ”pr_{Q}(\overline{C})”, forms a projective degeneration of C, in the sense of the previous lemma and remarks at the beginning of Section 1. Moreover, with {P_{1}, P_{2}} denoting the points of intersections (\overline{C} ∩ l) = (C ∩ l), the degeneration is asymptotic relative to the axis l.

Proof. The proof that this gives a degeneration, satisfying the previous lemma and remarks at the beginning of Section 1, is given in the previous lemma and the paper [6], using Lemma 8.1 to show that degree is preserved, and Lemma 8.3, to show that the genus may be preserved for a suitable choice of P (a birational projection will preserve the genus of the original curve). The limit curve ”pr_{Q}(\overline{C})” is not formally defined in terms of projections, however, is easily shown to be a union ⋃_{y∈(\overline{C}∩ω_{1})} l_{Qy} = ⋃_{y∈(C∩ω_{1})} l_{Qy} = l_{QP_{1}} ∪ l_{QP_{2}}, by the previous lemma. Again, the reader should look at the paper [6], Lemma 8.2. The asymptotic condition follows straightforwardly from the fact that the degeneration is linear, and both limit curve C_{∞} = pr_{Q}(C) and the initial curve C in the degeneration contains the points {P_{1}, P_{2}}, and the tangent lines {l_{P_{1}Q}, l_{P_{2}Q}}, which are fixed by each projection, being in the plane ω_{2}.

□

The above results correct the erroneous step of Severi’s conjecture, found in [20], as pointed out by Zariski in [28]. Zariski’s counterexample fails to apply to Severi’s cone construction, as it is not asymptotic. We will leave a closer examination of Severi’s proof to another paper.

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