A THEORY OF DIVISORS FOR ALGEBRAIC CURVES

TRISTRAM DE PIRO

Abstract. The purpose of this paper is two-fold. We first prove a series of results, concerned with the notion of Zariski multiplicity, mainly for non-singular algebraic curves. These results are required in [6], where, following Severi, we introduced the notion of the ”branch” of an algebraic curve. Secondly, we use results from [6], in order to develop a refined theory of g_{n}^{r} on an algebraic curve. This refinement depends critically on replacing the notion of a point with that of a ”branch”. We are then able to construct a theory of divisors, generalising the corresponding theory in the special case when the algebraic curve is non-singular, which is birationally invariant.

1. Introduction

In this paper, we use the same definition of an algebraic curve as in [6]. Namely, an algebraic curve C is a closed, irreducible subvariety of dimension 1 in P^{w}, for some w ≥ 1, where P^{w} denotes projective space of dimension w. We will often abbreviate the terminology of ”algebraic curve” to just ”curve”. The advantage of developing a birationally invariant theory of divisors for such curves depends mainly on the viewpoint of the ”Italian School” of algebraic geometry. Namely, that there are a number of benefits in studying the geometry of plane algebraic curves, (∗), and that any algebraic curve C is birational to a plane algebraic curve C′, (see, for example, Theorem 1.33 of [6]). It is not the purpose of this paper to discuss the question raised in (∗), leaving this point of view for another occasion. The results of this paper cover all characteristics of the underlying algebraically closed field L. However, we will make it clear when a result depends on the assumption that L has non-zero characteristic.

2. Smooth Curves

Before looking at this section, the reader is strongly advised to consult the paper [8] for relevant notation and terminology. In particular, the reader should be acquainted with the statement of Theorem 3.3 from [8]. We first recall the following theorem (which is Theorem 6.6 in [8]);

Thanks to Francesco Severi and the Lamb.

Theorem 2.1. Let hypotheses be as in Theorem 3.3 of [8], with the additional assumption that char(L) = 0 and F, D are smooth curves. Then the notions of Zariski multiplicity and algebraic multiplicity coincide.

As we need to refer to the proof of this result from [8] later in the paper, for the convenience of the reader, we repeat it below. The reader should, however, consult [8] for relevant notation.

Proof. As D has a non-constant meromorphic function, we can write D as a finite cover of P^{1}(L). As we have checked both algebraic multiplicity and Zariski multiplicity are multiplicative over composition (in [8]), a straightforward calculation shows that we need only check the notions agree for the branched finite cover π : F → P^{1}(L). (1)

Now consider this cover restricted to A^{1}, let x be the canonical coordinate with ord_{a}(π^{∗}(x)) = m, so we have that π^{∗}x = h^{m}u , for u a unit in O_{a} and h a uniformiser at a. (2)

As u is a unit and char(L) = 0, the equation z^{m} = u splits in the residue field of O_{a}^{∧}. By Hensel’s Lemma and Theorem 5.5 of [8], it is solvable in O_{a}^{∧}. By the definition of O_{a}^{∧}, we can find an etale morphism π : (U, b) → (F, a) containing such a solution in the local ring O_{b}. We may assume that U is irreducible and moreover, as π is etale, that U is smooth. (3)

Now we can embed U in a projective smooth curve F′ and, as F′ is smooth, extend the morphism π to a projective morphism from F′ to F. (4)

We claim that (ba) ∈ graph(π) ⊂ F′×F is unramified in the sense of Zariski structures. For this we need the following fact whose algebraic proof relies on the fact that etale morphisms are flat, see [8];

Fact 2.2. Any etale morphism can be locally presented in the form

V ⟶^{g} Spec((A[T]/f(T))_{d})
↓^{π}  ↓_{π′}
U ⟶^{h}  Spec(A)

where f(T) is a monic polynomial in A[T], f′(T) is invertible in (A[T]/f(T))_{d} and g, h are isomorphisms. (5)

Using Lemma 4.6 of [8] and the fact that the open set V is smooth, we may safely replace graph(π) by \overline{graph(π′)} ⊂ F′′ × F where F′′ is the projective closure of Spec((A[T]/f(T)), F is the projective closure of Spec(A) and \overline{graph(π′)} is the projective closure of graph(π′) and show that (g(b)a) is Zariski unramified. Note that over the open subset U = Spec(A) ⊂ F, \overline{graph(π′)} = Spec((A[T]/f(T) as this is closed in U × F′′. For ease of notation, we replace (g(b)a) by (ba). (6)

Suppose that f has degree n. Let σ_{1} … σ_{n} be the elementary symmetric functions in n variables T_{1}, … T_{n}. Consider the equations

σ_{1}(T_{1}, …, T_{n}) = a_{1}

…

σ_{n}(T_{1}, …, T_{n}) = a_{n} (*)

where a_{1}, … a_{n} are the coefficients of f with appropriate sign. These cut out a closed subscheme C ⊂ Spec(A[T_{1} … T_{N}]). Suppose (ba) ∈ graph(π′) = Spec(A[T]/f(T)) is ramified in the sense of Zariski structures, then I can find (a′b_{1}b_{2}) ∈ V_{abb} with (a′b_{1}),(a′b_{2}) ∈ Spec(A(T)/f(T)) and b_{1}, b_{2} distinct. Then complete (b_{1}b_{2}) to an n-tuple (b_{1}b_{2}c′_{1} … c′_{n−2}) corresponding to the roots of f over a′. The tuple (a′b_{1}b_{2}c′_{1} … c′_{n−2}) satisfies C, hence so does the specialisation (abbc_{1} … c_{n−2}). Then the tuple (bbc_{1} … c_{n−2}) satisfies (∗) with the coefficients evaluated at a. However such a solution is unique up to permutation and corresponds to the roots of f over a. This shows that f has a double root at (ab) and therefore f′(T)|_{ab} = 0. As (ab) lies inside Spec(A[T]/f(T))_{d}, this contradicts the fact that f′ is invertible in A[T]/f(T))_{d}. (7)

In (2) we may therefore assume that π^{∗}x = h^{m} for h a local uniformiser at a. Now we have the sequence of ring inclusions given by

L[x] → L[x, y]/(y^{m} − x) → R

x ↦ π^{∗}x, y ↦ h

where R is the coordinate ring of F in some affine neighborhood of a. It follows that we can factor our original map such that F is etale near a over the projective closure of y^{m} − x = 0. (8)

Again, repeating the argument from (4) to (7), we just need to check that the projective closure of y^{m} − x has multiplicity m at 0 considered as a cover of P^{1}(k̄). This is trival, let ϵ ∈ V_{0} be generic over

M,then as we are working in characteristic 0 we can find distinct ϵ_{1}, … ϵ_{m} in M_{∗} solving y^{m} = ϵ. By specialisation, each ϵ_{i} ∈ V_{0}. (9) □

The purpose of this section is essentially to find an analogous result to Theorem 2.1 when char(L) = p ≠ 0. An analogous result was given in [8], however, the proof was flawed. We correct this difficulty here. We obtained similar results, in [8], under different assumptions, by the straightforward method of counting points in the fibres. In this section, we need to use more sophisticated local methods, which will be explained below. We first make the following remark concerning the Frobenius morphism;

Remarks 2.3. Frobenius

Given a smooth curve C, defined over a field of characteristic p, with function field L(C), we let L(C)^{1/p} be the field obtained by extracting p^{th} roots of L(C) in some fixed algebraic closure. We denote by C_{p} the unique (up to isomorphism) smooth curve, having function field L(C)^{1/p}. Corresponding to the inclusion i : L(C) → L(C)^{1/p}, we obtain a morphism F rob : C_{p} → C, which, by some abuse of the standard terminology, (the standard terminology is L-linear Frobenius), we will refer to as Frobenius. Although L(C) and L(C)^{1/p} are clearly isomorphic as fields, they may not be isomorphic over L. Hence, C and C_{p} are not necessarily isomorphic curves. The Frobenius morphism may be explicitly realised as follows;

Let C be embedded in P^{n}, for some n, defined by the homogeneous polynomials \{f_{1}, …, f_{m}\}. Let C′ be the variety defined by \{\overline{f_{1}}, …, \overline{f_{m}}\},

A THEORY OF DIVISORS FOR ALGEBRAIC CURVES 5

where, for 1 ≤ j ≤ m, \overline{f_{j}} is the homogeneous polynomial obtained by applying inverse Frobenius to the coefficients. Then, by a straightforward calculation using Jacobians, C′ defines a smooth curve. The morphism Frobenius;

F r : P^{n} → P^{n} F r([X_{0} : …: X_{n}]) = [X_{0}^{p} : …: X_{n}^{p}] restricts to define a morphism F r : C′ → C. Let Rat_{k} denote the

rational functions of degree k on P^{n}. Then F r induces a map; F r^{∗} : Rat_{k} → Rat_{kp} by the formula; (F r^{∗}F)(X_{0}, …, X_{n}) = F (X_{0}^{p}, …, X_{n}^{p}) For a homogeneous polynomial f_{j} defining C, we have that; F r^{∗}(f_{j}) = (\overline{f}_{j})^{p} Hence, F r^{∗} restricts to define an L-linear map; F r^{∗} : L(C) → L(C′) One can also define a map; F r^{−1∗} : L[X_{0}, …, X_{n}] → L[X_{0}^{1/p}, …, X_{n}^{1/p}] by the formula; (F r^{−1∗}F)(X_{0}, …, X_{n}) = F (X_{0}^{1/p}, …, X_{n}^{1/p}) For a homogeneous polynomial \overline{f_{j}} defining C′, we have that; F r^{−1∗}(\overline{f_{j}}) = (f_{j})^{1/p} Hence, F r^{−1∗} restricts to define an L-linear isomorphism; F r^{−1∗} : L(C′) → L(C)^{1/p} (†)

We have that F r^{−1∗} ∘ F r^{∗} = Id, restricted to Rat_{k}, hence;

F r^{−1∗} ∘ F r^{∗} : L(C) → L(C′) → L(C)^{1/p} (††)

is the inclusion map. Using the fact that C_{p} and C′ are nonsingular projective curves, by (†) we obtain an isomorphism θ : C_{p} → C′. By (††), we have that;

F r ∘ θ = F rob : C_{p} → C

Hence, without loss of generality, we can identify the morphisms F r and the more abstractly defined morphism F rob.

We now make the following further remark.

Remarks 2.4. Given the hypotheses of Theorem 2.1, with the modification that char(L) = p ≠ 0, we define a point (ab) ∈ F to be wildly ramified if mult_{(ab)}^{alg}(F/D) is divisible by p. Theorem 2.1 holds excluding wildly ramified points, (∗). In order to see this, we first replace the argument (1), by showing that, for any given point a ∈ D, we can find a finite morphism f from D to P^{1}(L), such that f is etale in an open neighborhood of a;

As a is a non-singular, we can find a uniformising element t in the local ring O_{a,D} of D. Considering t as an element of the function field L(D), we obtain an embedding L(t) ⊂ L(D), which, as D is non-singular, determines a unique morphism f from D to P^{1}(L). Restricting the morphism to A^{1}(L) and letting x be the canonical coordinate, we have that f^{∗}(x) = t, hence ord_{a}(f^{∗}(x)) = 1. This shows that f is etale in an open neighborhood of a by Theorem 5.2 and Remarks 5.3 of [8]. (†)

As etale morphisms have multiplicity coprime to p, it is sufficient to check the result (∗) for a branched cover π : F → P^{1}(L). If a ∈ F is not wildly ramified for this cover, then we can follow through arguments (2) and (3) of Theorem 2.1. The argument from (4) to (8) is the same and we obtain the result of (9) again using the fact that m there is coprime to p. This proves the result (∗).

Theorem 2.1 also holds with the modification that char(L) = p ≠ 0 and the cover pr : F → D is seperable. However, the proof requires more sophisticated methods, which we consider below. We can, however, handle a special case by an elementary counting argument. First observe that we can replace the argument (1) by observing that there exists a seperable morphism f from D to P^{1}(L). This either follows from the argument (†) above or using the classical result that the function field L(D) admits a seperating transcendence basis over L, (see p27 of [2]). Hence, it is sufficient to check the result for a finite seperable cover π : F → P^{1}(L). By a classical result, (see Proposition 2.2, p300, of [2]), there exist finitely many ramification points, in particularly finitely many wild ramification points {a_{1}, …, a_{n}}, for the cover π. By the previous proof, we need only check the result of Theorem 2.1 for these finitely many points.

Special Case. a is a wild ramification point for the cover with the property that that there exist no other wild ramification points in the fibre π^{−1}(π(a)).

As both F and P^{1}(L) are non-singular, the finite morphism π is flat, by Lemma 5.11 of [8]. By a result in [3], (Corollary of Proposition 2, p218), we have that;

∑_{y∈π^{−1}(x)} mult_{y}^{alg}(F/P^{1}) is independent of x ∈ P^{1}(L), and equals the cardinality of a generic fibre.

By Lemma 4.3 of [8], a corresponding result also holds for Zariski multiplicities. Hence, by the result of the previous proof in this remark, the claim follows.

Unfortunately, one can have;

a is a wild ramification point for the cover with the property that there exist other wild ramification points {a_{1}, …, a_{r}}, distinct from a, in the fibre π^{−1}(π(a)).

It seems difficult to find any way of reducing this scenario to the special case. However, one can still use a local method, which is done in the following Theorem.

Theorem 2.5. Let hypotheses be as in Theorem 2.1, with the modification that char(L) = p ≠ 0 and the cover pr : F → D is seperable. Then the notions of Zariski multiplicty and algebraic multiplicity coincide.

Proof. By the previous remark, it is sufficient to consider the case when D is P^{1}(L). Let a ∈ F, such that, without loss of generality, pr(a) = 0 in the restriction of pr to A^{1}(L). As a is non-singular, we can find polynomials {f_{1}, …, f_{n−1}} in the variables {x_{1}, …, x_{n}} of an affine coordinate system A^{n}, such that a corresponds to the origin O of this system and F is defined locally by;

f_{1}(x_{1}, …, x_{n}) = … = f_{n−1}(x_{1}, …, x_{n}) = 0

with;

Jac(\frac{f_{1},...,f_{n−1}}{x_{2},...,x_{n}})|¯0 ≠ 0

We may then apply the implicit function theorem, (see for example p179 of [1]), in order to find power series {η_{1}, …, η_{n−1}}, in the variable t, with η_{j}(t) = 0, for 1 ≤ j ≤ n − 1, such that;

f_{j}(t, η_{1}(t), …, η_{n−1}(t)) = 0, for 1 ≤ j ≤ n − 1. (∗)

By (∗), we clearly have that the total transcendence degree of {t, η_{1}(t), …, η_{n−1}(t)} over L is equal to 1. Hence, we have that {η_{1}(t), …, η_{n−1}(t)} are algebraic over L(t). This implies, by the remarks at the beginning of Section 3 of [5], that they belong to the Henselisation of L[t]_{0}, hence they define functions on some etale cover (U, 0_{lift}) with coordinate ring L[t]^{ext} of (A^{1}, 0). We have the ring map;

\frac{L[x_{1},...,x_{n}]}{<f_{1},...,f_{n−1}>} →_{i} R = \frac{L[x_{1}]^{ext}[x_{2},...,x_{n}]}{<x_{2}−η(x_{1}),...,x_{n}−η_{n−1}(x_{1})>}

which corresponds to an etale cover (U′, a_{lift}) of (F, a). We also have an isomorphism;

R →_{γ} L[t]^{ext}; x_{1} ↦ t, x_{2} ↦ η_{1}(t), …, x_{n} ↦ η_{n−1}(t)

which corresponds to an isomorphism between (U, 0_{lift}) and (U′, a_{lift}). Now consider the composition;

θ : (U, 0_{lift}) → (U′, a_{lift}) → (F, a) →_{pr} (A^{1}, 0)

By the general method of [8], we can define both the algebraic and Zariski multiplicities of these covers. By Theorem 1.4 and Lemma 2.2 of [5], we have that;

A THEORY OF DIVISORS FOR ALGEBRAIC CURVES 9

Mult_{(0,a)}(F/D) = Mult_{(0,0_{lift})}(U/A^{1})

By Theorem 1.8 of [5], we also have that;

mult_{(0,a)}^{alg}(F/D) = mult_{(0,0_{lift})}^{alg}(U/A^{1})

Hence, the theorem is shown by proving that Zariski multiplicity and algebraic multiplicity coincide at (0, 0_{lift}) for the seperable cover θ. Suppose that the algebraic multiplicity is m, then, if t is the canonical coordinate for A^{1} at 0, we have that;

θ^{∗}t = t^{m}u(t) for a unit u(t) ∈ L[[t]] ∩ L(t)^{alg}

By the usual factoring argument, see (8) of Theorem 2.1, it is sufficent to check that the Zariski multiplicity of the seperable cover ϕ determined by;

L[s] → \frac{L[t]^{ext}[s]}{<t^{m}u(t)−s>}

is equal to m at (0, 0_{lift}) as well. This is done by the general method of Lemmas 4.5 and 4.6 of [5]. We apply Weierstrass preparation to t^{m}u(t) − s, see [1] for the power series version of this result, in order to obtain the factorisation;

t^{m}u(t) − s = u(t, s)(t^{m} + c_{1}(s)t^{m−1} + … + c_{m}(s)) = u(t, s)g(t, s) where c_{j}(s) ∈ L[[s]] ∩ L(s)^{alg}, c_{j}(s) = 0 for 1 ≤ j ≤ m and

u(t, s) ∈ L[[s, t]] ∩ L(s, t)^{alg} is a unit, see Lemma 3.2 of [5]. As is done in Lemma 4.6 of [5], we obtain the etale cover determined by;

\frac{L[t]^{ext}[s]}{<t^{m}u(t)−s>} → \frac{L[t,s]^{ext}}{<u(t,s)g(t,s)>} By the argument there, it is sufficient to determine when the Weier-

strass factor g(t, s) determines a generically reduced cover. Using the method of resultants in Lemma 4.5 of [5], this occurs if and only if \frac{∂g}{∂t} is not identically zero. If \frac{∂g}{∂t} is identically zero, we obtain the factorisation g(t, s) = h(t^{p}, s). This clearly implies that the original cover ϕ is inseperable, which is a contradiction. The theorem is then proved.

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We now have; Theorem 2.6. Let hypotheses be as in Theorem 2.1, with the modification that char(L) = p ≠ 0. If e denotes the Zariski multiplicity and d the algebraic multiplicity at a ∈ F, then d = ep^{n} and π factors as F →_{h} F′ →_{g} D with h = F rob^{n} and g having algebraic multiplicity e at h(a).

Proof. As in Theorem 6.3 of [8], we can factor π into a purely inseperable morphism h : F → F′ and a seperable morphism g : F′ → D with F′ a smooth projective curve. We then have a corresponding sequence of field extensions L(D) ⊂ L(F′) ⊂ L(F), with L(F) a purely inseperable extension of L(F′). As L(F) is a purely inseperable field extension of L(F′), it has degree p^{n} for some n ≥ 1. Hence, L(F) = L(F′)^{1/p^{n}} and we may, without loss of generality, assume that h = F rob^{n}, see also Proposition 2.5 (p302) of [2]. By the previous theorem, the notions of Zariski multiplicity and algebraic multiplicity coincide for the morphism g. By Remarks 2.3, the Frobenius morphism F rob may be identified with F r, without effecting Zariski or algebraic multiplicities. Clearly, F r is a bijection on points, hence it is Zariski unramified. F r has algebraic multiplicity p everywhere, as, for any point x ∈ F′, we can choose a local uniformiser t at x such that F r^{∗}(t) = t^{p}. It follows that h has algebraic multiplicity p^{n} everywhere and is Zariski unramified. The result now follows immediately from Lemma 4.5 and Remarks 5.7 of [8].

□

We now give a local version of Theorem 2.1 in the general case of algebraic curves over a field L with char(L) = 0 and find an analogous version of Theorem 2.5, in the case when char(L) = p ≠ 0.

Theorem 2.7. Let hypotheses be as in Theorem 3.3 of [8], with the additional assumption that char(L) = 0 and D is a smooth curve. Let pr be the projection map of F onto D. Then, if (ab) ∈ F is non-singular;

Mult_{ab}(F/D) = mult_{ab}^{alg}(F/D)

that is Zariski multiplicity and algebraic multiplicity coincide. In particular, the cover (F/D) is Zariski unramified at (ab) iff there exists an open U ⊂ F, containing (ab), such that pr : U → D is etale.

Proof. For the first part of the theorem, we follow the proof of Theorem 2.1, the difference between the hypotheses there is that we do not assume that F is smooth. Using the fact that D is smooth and the result of Theorem 2.1, we may, without loss of generality, assume that D = P^{1}(L). Now, one can follow through the proof of Theorem 2.1, using the fact that (ab) is non-singular, in order to obtain the result. One should make the modification that Zariski multiplicity is well defined for any finite cover F′ → F at (abc) lying over (ab). This follows from an easy extension of Theorem 3.3 (in [8]), to show that a nonsingular open subvariety of an irreducible projective variety of dimension 1 is presmooth (see [4]). For the second part of the theorem, suppose that there exists an open U ⊂ F, containing (ab), such that pr : U → D is etale. As (ab) is non-singular, we may assume that U defines a nonsingular open subvariety of F. Following the argument of Theorem 2.1, from the end of (4) to the end of (7), we obtain that the cover (F/D) is Zariski unramified at (ab). For the converse, assume that the cover is Zariski unramified at (ab). By Theorem 5.2, Remarks 5.3 of [8] and the fact that (ab) is non-singular, it is sufficient to prove that d(pr) : (m_{(ab)}/m_{(ab)}^{2})^{∗} → (m_{a}/m_{a}^{2})^{∗} is an isomorphism. Equivalently, we need to show that the algebraic multiplicity mult_{(ab)}^{alg}(F/D) of pr at (ab) ∈ F equals 1. This follows from the first part of the theorem.

□

Theorem 2.8. Let hypotheses be as in Theorem 3.3 of [8], with the additional assumption that char(L) = p ≠ 0, D is a smooth curve and the projection map pr of F onto D is seperable. Then, if (ab) ∈ F is non-singular;

Mult_{ab}(F/D) = mult_{ab}^{alg}(F/D)

that is Zariski multiplicity and algebraic multiplicity coincide. In particular, the cover (F/D) is Zariski unramified at (ab) iff there exists an open U ⊂ F, containing (ab), such that pr : U → D is etale.

Proof. Here, the hypotheses are the same as Theorem 2.5, with the modification that we do not assume F is smooth. The proof is similar to the previous theorem. By Remarks 2.4, we can assume that D = P^{1}(L). Using the fact that (ab) is non-singular, one can either follow through the proof of Theorem 2.1, if (ab) is not wildly ramified for the cover, or one can use the method in Theorem 2.5, if (ab) is wildly ramified for the cover. For the second part, one can use the same reasoning as in the previous theorem.

□

Remarks 2.9. This last result is required for the proof of Lemma 2.10 from [6] under suitable assumptions, when char(L) = p ≠ 0. The reader should consult the final section on Frobenius from the paper [6].

We finish this section with the following result;

Theorem 2.10. Let G(X, Y) = 0 define an irreducible plane algebraic curve C, with a non-singular point at (0, 0). Let (T, η(T)) be a power series representation of this point. Then, for any plane, possibly reduced, algebraic curve F (X, Y) = 0 passing through (0, 0);

F (T, η(T)) ≡ 0 iff F contains C as a component.

Otherwise, I(G, F, (0, 0)) = ord_{T} F (T, η(T)).

Proof. The proof partly uses the methods of [5]. For the first part, note that if F contains C as a component, then by the Nullstellenstatz, there exists H(X, Y) such that F (X, Y) = H(X, Y)G(X, Y). It then follows trivially that F (T, η(T)) ≡ 0. For the converse direction, suppose that F (T, η(T)) ≡ 0. As in Lemma 4.17 of [5], we may interpret the equation Y − η(X) as defining a curve C_{1} on some etale extension i : (A_{et}^{2}, (00)^{lift}) → (A^{2}, (00)) such that i(C_{1}) ⊂ C. The vanishing of F (X, Y) on C_{1} then implies that F intersects C in an open dense subset. Therefore, as both F and C define Zariski closed sets, F must contain C as a component. For the second part of the theorem, we may therefore assume that F has finite intersection with C and ord_{T} F (T, η(T)) is defined. Suppose that F (X, Y) has degree d and consider F as part of the family of degree d curves Q_{d}. Without loss of generality, we may suppose that F (X, Y) = H(X, Y, v̄^{0}) where, for v̄ ∈ P ar_{Q_{d}}, H(X, Y, v̄) defines an algebraic curve of degree d. Similarily, we can write G(X, Y) in the form G(X, Y, ū^{0}) for some non-varying constant ū^{0}. As in Lemma 4.17 of [5], we have the sequence of maps;

L[v̄] → \frac{L[X,Y][v̄]}{<G(X,Y,ū_{0}),H(X,Y,v̄)>} → \frac{L[X]^{ext}[Y][v̄]}{<Y −η(X),H(X,Y,v̄)>}

which corresponds to a sequence of finite covers;

F_{1} → F′(ū^{0}, V) → Spec(L[v̄])

One checks that the left hand morphism is etale at (v̄^{0}, (00)^{lift}), by direct calculation. We use the fact that F is non-singular at (00), therefore the completion of the local rings \frac{L[X,Y]}{<G(X,Y,ū_{0})>}_{(00)} and \frac{L[X]^{ext}[Y]}{<Y −η(X)>}_{(00)} are in both cases equal to the formal power series ring L[[X]].

We now compute the Zariski multiplicity of the cover F_{1} → Spec(L[v̄]) at (v̄^{0}, (00)^{lift}) (∗). We are given the formal power series H(X, η(X), v̄) ∈ L[[X, v̄]]. Let d = ord_{X}H(X, η(X), v̄_{0}). Then, by Weierstrass preparation in several variables, see [1], we can find H_{1}(X, v̄) and U(X, v̄) in L[[X, v̄]] such that;

H(X, η(X), v̄) = H_{1}(X, v̄)U(X, v̄)

and U(0, v̄_{0}) ≠ 0 and

H_{1}(X, v̄) = X^{d} + c_{1}(v̄)X^{d−1} + … c_{d}(v̄)

with c_{j}(v̄_{0}) = 0 for 1 ≤ j ≤ d. Now use the proofs of Lemma 4.5 and 4.6 from [5] and the fact that the cover;

Spec(H_{1}(X, v̄)) → Spec(L[v̄])

is generically reduced to show the Zariski multiplicity of the cover (∗) is exactly d. This proves that the Zariski multiplicity of the cover;

F′(ū^{0}, V) → Spec(L[v̄])

at ((0, 0), v̄^{0}) is exactly d as well. By the general result of the paper [5], that;

I(C_{}_{ū^{0}}, C_{}_{v̄^{0}}, (00)) = RightMult_{(00)}(C_{}_{ū^{0}}, C_{}_{v̄^{0}})

when C_{}_{ū^{0}} defines a reduced algebraic curve, the result of the theorem follows.

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Remarks 2.11. This last Theorem was required in the proof of Theorem 6.1 of [6]. It is also required in the proof of Remarks 4.8 below.

3. A refined theory of g_{n}^{r}

The purpose of this section is to refine the general theory of g_{n}^{r}, given in [6], in order to take into account the notion of a branch for a projective algebraic curve. We will rely heavily on results proved in [6]. We also refer the reader there for the relevant notation. We will make no assumptions on the characteristic of the base field L. As usual, by an algebraic curve, we always mean a projective irreducible variety of

14 TRISTRAM DE PIRO

dimension 1. Definition 3.1. Let C ⊂ P^{w} be a projective algebraic curve of degree d and let Σ be a linear system of dimension R, contained in the space of algebraic forms of degree e on P^{w}. Let ϕ_{λ} belong to Σ, having finite intersection with C. Then, if p ∈ C ∩ ϕ_{λ} and γ_{p} is a branch centred at p, we define;

I_{p}(C, ϕ_{λ}) = I_{italian}(p, C, ϕ_{λ}) I_{p}^{Σ}(C, ϕ_{λ}) = I_{italian}^{Σ}(p, C, ϕ_{λ}) I_{p}^{Σ,mobile}(C, ϕ_{λ}) = I_{italian}^{Σ,mobile}(p, C, ϕ_{λ}) I_{γ_{p}}(C, ϕ_{λ}) = I_{italian}(p, γ_{p}, C, ϕ_{λ}) I_{γ_{p}}^{Σ}(C, ϕ_{λ}) = I_{italian}^{Σ}(p, γ_{p}, C, ϕ_{λ}) I_{γ_{p}}^{Σ,mobile}(C, ϕ_{λ}) = I_{italian}^{Σ,mobile}(p, γ_{p}, C, ϕ_{λ}) where I_{italian} was defined in [6]. It follows that, as λ varies in P ar_{Σ}, we obtain a series of weighted

sets; W_{λ} = {n_{γ_{p_{1}}^{1}}, …, n_{γ_{p_{1}}^{n_{1}}}, …, n_{γ_{pm}^{1}}, …, n_{γ_{pm}^{nm}}} where; {p_{1}, …, p_{i}, …, p_{m}} = C ∩ ϕ_{λ}, for 1 ≤ i ≤ m, {γ_{p_{i}}^{1}, …, γ_{p_{i}}^{j(i)}, …, γ_{p_{i}}^{n_{i}}}, for 1 ≤ j(i) ≤ n_{i}, consists of the branches of

C centred at p_{i} and I_{γ_{p_{i}}^{j(i)}}(C, ϕ_{λ}) = n_{γ_{p_{i}}^{j(i)}} By the branched version of the Hyperspatial Bezout Theorem, see

[6], the total weight of any of these sets, which we will occasionally abbreviate by C ⊓ ϕ_{λ}, is always equal to de. Let r be the least integer such that every weighted set W_{λ} is defined by a linear subsystem Σ′ ⊂ Σ of dimension r.

Definition 3.2. We define;

Series(Σ) = {W_{λ} : λ ∈ P ar_{Σ}}

dimension(Series(Σ)) = r

order(Series(Σ)) = de

We then claim the following;

Theorem 3.3. .

(i). r ≤ R, with equality iff every weighted set W_{λ} of the series is cut out by a single form of Σ.

(ii). r ≨ R iff there exists a form ϕ_{λ} in Σ, containing all of C.

Proof. We first show the equivalence of (i) and (ii). Suppose that (i) holds and r ≨ R. Then, we can find a weighted set W and distinct elements {λ_{1}, λ_{2}} of P ar_{Σ} such that W = W_{λ_{1}} = W_{λ_{2}}. Let {ϕ_{λ_{1}}, ϕ_{λ_{2}}} be the corresponding algebraic forms of Σ and consider the pencil Σ_{1} ⊂ Σ defined by these forms. We claim that;

W = C ⊓ (μ_{1}ϕ_{λ_{1}} + μ_{2}ϕ_{λ_{2}}), for [μ_{1} : μ_{2}] ∈ P^{1} (∗)

This follows immediately from the results in [6] that the condition of multiplicity at a branch is linear and the branched version of the Hyperspatial Bezout Theorem. Now choose a point p ∈ C, which is not a base point for any of the branches in W. Then, the condition that an algebraic form ϕ_{λ} passes through p defines a hyperplane condition on P ar_{e}, hence, intersects P ar_{Σ_{1}} in a point. Let ϕ_{λ_{0}} be the algebraic form in Σ_{1} defined by this parameter. Then, by (∗), we have that;

W ∪ {p} ⊆ C ⊓ ϕ_{λ_{0}}

Hence, the total multiplicity of intersection of ϕ_{λ_{0}} with C is at least equal to de + 1. By the branched version of the Hyperspatial Bezout Theorem, C must be contained in ϕ_{λ_{0}}. Conversely, suppose that (i) holds and there exists a form ϕ_{λ_{0}} in Σ containing all of C. Let W be cut out by ϕ_{λ_{1}} and consider the pencil Σ_{1} ⊂ Σ generated by {ϕ_{λ_{0}}, ϕ_{λ_{1}}}. By the same argument as above, we can find ϕ_{λ_{2}} in Σ_{1}, distinct from ϕ_{λ_{1}}, which also cuts out W. Hence, by (i), we must have that r ⪇ R. Therefore, (ii) holds.

The argument that (ii) implies (i) is similar.

We now prove that (ii) holds. Using the Hyperspatial Bezout Theorem, the condition on P ar_{Σ} that a form ϕ_{λ} contains C is linear. Let H be the linear subsystem of Σ, consisting of forms containing C and let h = dim(H). Let K ⊂ Σ be a maximal linear subsystem, having finite intersection with C. Then K has no form in common with H and dim(K) = R − h − 1. We claim that every weighted set in Series(Σ) is cut out by a unique form from K. For suppose that W = C ⊓ϕ_{λ} is such a weighted set and consider the linear system defined by < H, ϕ_{λ} >. If ϕ_{μ} belongs to this system and has finite intersection with C, then clearly (C ∩ ϕ_{λ}) = (C ∩ ϕ_{μ}). Using linearity of multiplicity at a branch and the Hyperspatial Bezout Theorem again (by convention, a form containing C has infinite multiplicity at a branch), we must have that (C ⊓ ϕ_{λ}) = (C ⊓ ϕ_{μ}). Now consider K∩ < H, ϕ_{λ} >. We have that;

codim(K∩ < H, ϕ_{λ} >) ≤ codim(K) + codim(< H, ϕ_{λ} >) . = (h + 1) + (R − (h + 1)) = R.

Hence, dim(K∩ < H, ϕ_{λ} >) ≥ 0. We can, therefore, find a form ϕ_{μ} belonging to K such that W = (C ⊓ ϕ_{μ}). We need to show that ϕ_{μ} is the unique form in K defining W. This follows by the argument given above. It follows immediately that r = dim(K) = R − h − 1. Hence, r ⪇ R iff h ≥ 0. Therefore, (ii) is shown.

□

Using this theorem, we give a more refined definition of a g_{n}^{r}.

Definition 3.4. Let C ⊂ P^{w} be a projective algebraic curve. By a g_{n}^{r} on C, we mean the collection of weighted sets, without repetitions, defined by Series(Σ) for some linear system Σ, such that r = dimension(Series(Σ)) and n = order(Series(Σ)). If a branch γ_{p}^{j} appears with multiplicity at least s in every weighted set of a g_{n}^{r}, as just defined, then we allow the possibility of removing some multiplicity contribution s′ ≤ s from each weighted set and adjusting n to n′ = n − s′.

Remarks 3.5. The reader should observe carefully that a g_{n}^{r} is defined independently of a particular linear system. However, by the previous theorem, for any g_{n}^{r}, there exists a g_{n′}^{r} with n ≤ n′ such that the following property holds. The g_{n′}^{r} is defined by a linear system of dimension r, having finite intersection with C, such that each there is a bijection between the weighted sets W in the g_{n′}^{r} and the W_{λ} in Series(Σ). The original g_{n}^{r} is obtained from the g_{n′}^{r} by removing some fixed branch contribution.

We now reformulate the results of Section 2 and Section 5 in [6] for this new definition of a g_{n}^{r}. In order to do this, we require the following definition;

Definition 3.6. Suppose that C ⊂ P^{w}(L) is a projective algebraic curve and C^{ext} ⊂ P^{w}(K) is its non-standard model. Let a g_{n}^{r} be given on C, defined by a linear system Σ after removing some fixed branch contribution. We define the extension g_{n}^{r,ext} of the g_{n}^{r} to the nonstandard model C^{ext} to be the collection of weighted sets, without repetitions, defined by Series(Σ) on C^{ext}, after removing the same fixed point contribution. Note that, by definability of multiplicity at a branch, see Theorem 6.5 of [6], if γ_{p}^{j} is a branch of C and;

I_{italian}(p, γ_{p}^{j}, C, ϕ_{λ}) ≥ k, (λ ∈ P ar_{Σ(L)})

then;

I_{italian}(p, γ_{p}^{j}, C, ϕ_{λ}) ≥ k, (λ ∈ P ar_{Σ(K)})

Hence, it is possible to remove the same fixed point contribution of Series(Σ) on C^{ext}. See also the proof of Lemma 3.7.

It is a remarkable fact that, after introducing the notion of a branch, the definition is independent of the particular linear system Σ. This is the content of the following lemma;

Lemma 3.7. The previous definition is independent of the particular choice of linear system Σ defining the g_{n}^{r}.

Proof. We divide the proof into the following cases;

Case 1. Σ ⊂ Σ′;

By the proof of Theorem 3.3, we can find a linear system Σ_{0} ⊂ Σ ⊂ Σ′ of dimension r, having finite intersection with C, such that the g_{n}^{r} is defined by removing some fixed contribution from Σ_{0}. Here, we have also used the fact that the base point contributions (at a branch) of {Σ_{0}, Σ, Σ′} are the same. Again, by Theorem 3.3, if W_{λ′} is a weighted set defined by Σ′ on C^{ext}, then it appears as a weighted set V_{λ′′} defined by Σ_{0} on C^{ext}. Hence, it appears as a weighted set V_{λ′′} defined by Σ on C^{ext}. By the converse argument and Remarks 3.5 on base branch contributions, the proof is shown.

Case 2. Σ are Σ′ are both linear systems of dimension r, having finite intersection with C, such that degree(Σ) = degree(Σ′) = n;

By Theorem 3.3, every weighted set W in the g_{n}^{r} is defined uniquely by weighted sets W_{λ_{1}} and V_{λ_{2}} in Series(Σ_{1}) and Series(Σ_{2}) respectively. Let (C^{ns}, Φ^{ns}) be a non-singular model of C. Using the method of Section 5 in [6] to avoid the technical problem of presentations of Φ^{ns} and base point contributions, we may, without loss of generality, assume that there exist finite covers W_{1} ⊂ P ar_{Σ} × C^{ns} and W_{2} ⊂ P ar_{Σ′} × C^{ns} such that;

j_{k,Σ}(λ, p_{j}) ≡ Mult_{(W_{1}/P ar_{Σ})}(λ, p_{j}) ≥ k iff I_{italian}(p, γ_{p}^{j}, C, ϕ_{λ}) ≥ k

j_{k,Σ′}(λ′, p_{j}) ≡ Mult_{(W_{2}/P ar_{Σ′})}(λ′, p_{j}) ≥ k iff I_{italian}(p, γ_{p}^{j}, C, ψ_{λ′}) ≥ k

Then consider the sentences;

(∀λ ∈ P ar_{Σ})(∃!λ′ ∈ P ar_{Σ′})∀x ∈ C^{ns}[⋀_{k=1}^{n}(j_{k}(λ, x) ↔ j_{k}(λ′, x))]

(∀λ′ ∈ P ar_{Σ})(∃!λ ∈ P ar_{Σ})∀x ∈ C^{ns}[⋀_{k=1}^{n}(j_{k}(λ′, x) ↔ j_{k}(λ, x))] (*)

in the language of < P^{1}(L), C_{i} >, considered as a Zariski structure with predicates {C_{i}} for Zariski closed subsets defined over L, (see [4]). We have, again by results of [4] or [7], that < P^{1}(L), C_{i} >≺< P^{1}(K), C_{i} >, for the nonstandard model P (K) of P (L). It follows immediately from the algebraic definition of j_{k} in [4], that, for any weighted set W_{λ′_{1}} defined by Series(Σ) on C^{ext}, there exists a unique weighted set V_{λ′_{2}} defined by Series(Σ′) on C^{ext} such that W_{λ′_{1}} = V_{λ′_{2}}, and conversely. Hence, the proof is shown.

Case 3. Σ are Σ′ are both linear systems of dimension r, having finite intersection with C;

Let n_{1} = degree(Σ) and n_{2} = degree(Σ′). Then the original g_{n}^{r} is obtained from Series(Σ), by removing a fixed point contribution of multiplicity n_{1} − n, and, is obtained from Series(Σ′), by removing a fixed point contribution of multiplicity n_{2} − n. We now imitate the proof of Case 2, with the slight modification that, in the construction of the sentences given by (∗), we make an adjustment of the multiplicity statement at the finite number of branches where a fixed point contribution has been removed. The details are left to the reader. □

Now, using Definition 3.6, we construct a specialisation operator sp : g_{n}^{r,ext} → g_{n}^{r}. We first require the following simple lemma;

Lemma 3.8. Let C ⊂ P^{w}(L) be a projective algebraic curve and let C^{ext} ⊂ P^{w}(K) be its nonstandard model. Let p′ ∈ C^{ext} be a nonsingular point, with specialisation p ∈ C. Then there exists a unique branch γ_{p}^{j} such that p′ ∈ γ_{p}^{j}.

Proof. We may assume that p′ ≠ p, otherwise p would be non-singular and, by Lemma 5.4 of [6], would be the origin of a single branch γ_{p}. Let (C^{ns}, Φ) be a non-singular model of C, then p′ must belong to the canonical set V_{[Φ]}, hence there exists a unique p′′ ∈ C^{ns} such that Φ(p′′) = p′. By properties of specialisations, p′′ ∈ C^{ns} ∩ V_{p_{j}} for some p_{j} ∈ Γ_{[Φ]}(x, p). Hence, by definition of a branch given in Definition 5.15 of [6], we must have that p′ ∈ γ_{p}^{j}. The uniqueness statement follows as well. □

We now make the following definition;

Definition 3.9. Let C ⊂ P^{w}(L) be a projective algebraic curve and let C^{ext} ⊂ P^{w}(K) be its non-standard model. Given a g_{n}^{r} on C with extension g_{n}^{r,ext} on C^{ext}, we define the specialisation operator;

sp : g_{n}^{r,ext} → g_{n}^{r}

by;

sp(γ_{p′}) = γ_{p}^{j}, for p′ ∈ NonSing(C^{ext}) and γ_{p}^{j} as in Lemma 3.8. sp(γ_{p}^{j}) = γ_{p}^{j}, for p ∈ Sing(C^{ext}) = Sing(C) and {γ_{p}^{1}, …, γ_{p}^{j}, …, γ_{p}^{s}}

enumerating the branches at p.

sp(n_{1}γ_{p_{1}}^{j_{1}} + … + n_{r}γ_{p_{r}}^{j_{r}}) = n_{1}sp(γ_{p_{1}}^{j_{1}}) + … + n_{r}sp(γ_{p_{r}}^{j_{r}}),

for a linear combination of branches with n_{1} + … + n_{r} = n

It is also a remarkable fact that, after introducing the notion of a branch, the specialisation operator sp is well defined. This is the content of the following lemma;

Lemma 3.10. Let hypotheses be as in the previous definition, then, if W is a weighted set belonging to g_{n}^{r,ext}, its specialisation sp(W) belongs to g_{n}^{r}.

Proof. We may assume that there exists a linear system Σ, having finite intersection with C, such that dimension(Σ) = r and degree(Σ) = n_{1}, with the g_{n}^{r} and g_{n}^{r,ext} both defined by Series(Σ), after removing some fixed branch contribution W_{0} of multiplicity n_{1} − n. Let W be a weighted set of the g_{n}^{r,ext}, then W ∪ W_{0} = (C ⊓ ϕ_{λ′}), for some unique λ′ ∈ P ar_{Σ}. We claim that sp(W ∪ W_{0}) = C ⊓ ϕ_{λ}, for the specialisation λ ∈ P ar_{Σ} of λ′ (∗). As sp(W_{0}) = W_{0}, it then follows immediately from linearity of sp, that sp(W) belongs to the g_{n}^{r} as required. We now show (∗). Let p ∈ C and let γ_{p} be a branch centred at p. By γ_{p}^{ext}, we mean the branch at p, where p is considered as an element of C^{ext}. We now claim that;

I_{γ_{p}}(C, ϕ_{λ}) = I_{γ_{p}^{ext}}(C, ϕ_{λ′}) + ∑_{p′∈(γ_{p}\p)} I_{γ_{p′}^{ext}}(C, ϕ_{λ′}) (∗∗)

Let (C^{ns}, Φ) ⊂ P^{w′}(L) be a non-singular model of C, such that γ_{p} corresponds to C^{ns,ext} ∩ V_{q}, where q ∈ Γ_{[Φ]}(x, p) and V_{q} is defined relative to the specialisation from P (K) to P (L). Let C^{ns,ext,ext} ⊂ P^{w′}(K′) be a non-standard model of C^{ns,ext}, such that γ_{q}^{ext} corresponds to C^{ns,ext,ext} ∩ V_{q}, where V_{q} is defined relative to the specialisation from P (K′) to P (K). Then, for p′ ∈ (γ_{p} \ p), we can find q′ ∈ V_{q} ∩ C^{ns,ext} such that γ_{p′} corresponds to V_{q′} ∩ C^{ns,ext,ext}. We may choose a suitable presentation Φ_{Σ_{1}} of Φ, such that Base(Σ_{1}) is disjoint from Γ_{[Φ]}(x, p), and, therefore, disjoint from Γ_{[Φ]}(x, p′), for p′ ∈ (γ_{p} \ p). Let \{\overline{ϕ_{λ}}\} denote the lifted family of on C^{ns} from the presentation Φ_{Σ′}. In this case, we have, by results of [6], that;

A THEORY OF DIVISORS FOR ALGEBRAIC CURVES 21

I_{γ_{p}}(C, ϕ_{λ}) = I_{q}(C^{ns}, \overline{ϕ_{λ}}) I_{γ_{p}^{ext}}(C, ϕ_{λ′}) = I_{q}(C^{ns}, \overline{ϕ_{λ′}}) I_{γ_{p′}^{ext}}(C, ϕ_{λ′}) = I_{q′}(C^{ns}, \overline{ϕ_{λ′}}) (1) By summability of specialisation, see [6] and [5]; I_{q}(C^{ns}, \overline{ϕ_{λ}}) = I_{q}(C^{ns}, \overline{ϕ_{λ′}}) + ∑_{q′∈C^{ns}∩(V_{q}\q)} I_{q′}(C^{ns}, \overline{ϕ_{λ′}}) (2) Combining (1) and (2), the result (∗∗) follows, as required. Now,

suppose that a branch γ_{p} occurs with non-trivial multiplicity in sp(C ⊓ ϕ_{λ′}). By Definition 3.9, the contribution must come from either I_{γ_{p}^{ext}}(C, ϕ_{λ′}) or I_{γ_{p′}^{ext}}(C, ϕ_{λ′}), for some p′ ∈ (γ_{p} \ p). Applying sp to (∗∗), one sees that the branch γ_{p} occurs with multiplicity I_{γ_{p}}(C, ϕ_{λ}). It follows that sp(C ⊓ ϕ_{λ′}) = C ⊓ ϕ_{λ}, hence (∗) is shown. The lemma then follows.

□

We can now reformulate the results of Section 2 and Section 5 of [6] in the language of this refined theory of g_{n}^{r}. We first make the following definition;

Definition 3.11. Let C ⊂ P^{w} be a projective algebraic curve and let a g_{n}^{r} be given on C. Let W be a weighted set in this g_{n}^{r} or its extension g_{n}^{r,ext} and let γ_{p} be a branch centred at p. Then we say that;

γ_{p} is s-fold (s-plo) for W if it appears with multiplicity at least s. γ_{p} is multiple for W if it appears with multiplicity at least 2. γ_{p} is simple for W if it is not multiple. γ_{p} is counted (contato) s-times in W if it appears with multiplicity

exactly s.

γ_{p} is a base branch of the g_{n}^{r} if it appears in every weighted set. γ_{p} is s-fold for the g_{n}^{r} if it is s-fold in W for every weighted set W

of the g_{n}^{r}.

γ_{p} is counted s-times for the g_{n}^{r} if it is s-fold for the g_{n}^{r} and is counted s-times in some weighted set W of the g_{n}^{r}.

We then have the following;

Theorem 3.12. Local Behaviour of a g_{n}^{r}

Let C be a projective algebraic curve and let a g_{n}^{r} be given on C. Let γ_{p} be a branch centred at p, such that γ_{p} is counted s-times for the g_{n}^{r}. If γ_{p} is counted t times in a given weighted set W, then there exists a weighted set W′ in g_{n}^{r,ext} such that sp(W′) = W and sp^{−1}(tγ_{p}) consists of the branch γ_{p} counted s-times and t − s other distinct branches {γ_{p_{1}}, …, γ_{p_{t−s}}}, each counted once in W′.

Proof. Without loss of generality, we may assume that the g_{n}^{r} is defined by a linear system Σ of dimension r, having finite intersection with C. Let W be the weighted set defined by ϕ_{λ} in Σ. Suppose that s = 0, then γ_{p} is not a base branch for Σ. Hence, by Lemma 5.25 of [6], we can find λ′ ∈ V_{λ}, generic in P ar_{Σ}, and distinct {p_{1}, …, p_{t}} = C^{ext}∩ϕ_{λ′} ∩(γ_{p} \ p) such that the intersections at these points are transverse. Let W′ be the weighted set defined by ϕ_{λ′} in g_{n}^{r,ext}. By the proof of (∗) in Lemma 3.10, we have that sp(W′) = W. By the construction of sp in Definition 3.9, we have that sp^{−1}(tγ_{p}) consists of the distinct branches {γ_{p_{1}}, …, γ_{p_{t}}}, each counted once in W′. If s ≥ 1, then γ_{p} is a base branch for Σ. By Lemma 5.27 of [6], we have that I_{italian}^{Σ,mobile}(p, γ_{p}, C, ϕ_{λ}) = t − s. The result then follows by application of Lemma 5.28 in [6] and the argument given above. □

We now note the following;

Lemma 3.13. Let a g_{n}^{r} be given on a projective algebraic curve C. Let W_{0} be any weighted set on C with total multiplicity n′. Then the collection of weighted sets given by {W ∪ W_{0}} for the weighted sets W in the g_{n}^{r} defines a g_{n+n′}^{r}.

Proof. Let the original g_{n}^{r} be obtained from a linear system Σ of dimension r and degree n′′, having finite intersection with C, after removing some fixed branch contribution J of total multiplicity n′′ − n. Let {ϕ_{0}, …, ϕ_{r}} be a basis for Σ and let {n_{1}γ_{p_{1}}^{j_{1}}, …, n_{m}γ_{p_{m}}^{j_{m}}} be the branches appearing in W_{0} with total multiplicity n_{1} + … + n_{m} = n′ (†). Let {H_{1}, …, H_{m}} be hyperplanes passing through the points {p_{1}, …, p_{m}} and let G be the algebraic form of degree n′ defined by H_{1}^{n_{1}} ▪ … ▪ H_{m}^{n_{m}}. Let Σ′ be the linear system of dimension r defined by the basis {G ▪ ϕ_{0}, …, G ▪ ϕ_{r}}. As we may assume that C is not contained in any hyperplane section, Σ′ has finite intersection with C. We claim that g_{n′′}^{r}(Σ) ⊂ g_{n′′+n′deg(C)}^{r}(Σ′), in the sense that every weighted set W_{λ} defined by g_{n′′+n′deg(C)}^{r}(Σ′) is obtained from the corresponding V_{λ} in g_{n′′}^{r}(Σ) by adding a fixed weighted set W_{1} ⊃ W_{0} of total multiplicity n′deg(C) (∗). The proof then follows as we can recover the original g_{n}^{r} by removing the fixed branch contribution J ∪ (W_{1} \ W_{0}) from g_{n′′+n′deg(C)}^{r}(Σ′). In order to show (∗), let W_{1} be the weighted set defined by C ⊓ G. By the branched version of the Hyperspatial Bezout Theorem, see Theorem 5.13 of [6], this has total multiplicity n′deg(C). We claim that W_{0} ⊂ W_{1} (∗∗). Let γ_{p}^{j} be a branch appearing in (†) with multiplicity s. By construction, we can factor G as H^{s} ▪ R, where H is a hyperplane passing through s. We need to show that;

I_{γ_{p}^{j}}(C, H^{s} ▪ R) ≥ s

or equivalently,

I_{p_{j}}(C^{ns}, \overline{H^{s} ▪ R}) = I_{p_{j}}(C^{ns}, \overline{H}^{s} ▪ \overline{R}) ≥ s

for a suitable presentation C^{ns} of a non-singular model of C, see Lemma 5.12 of [6], where we have used the ”lifted” form notation there. Using the method of conic projections, see section 4 of [6], we can find a plane projective curve C′ birational to C^{ns}, such that the point p_{j} corresponds to a non-singular point q of C′ and;

I_{p_{j}}(C^{ns}, \overline{H}^{s} ▪ \overline{R}) = I_{q}(C′, \overline{\overline{H}^{s} ▪ \overline{R}}) = I_{q}(C′, \overline{\overline{H}}^{s} ▪ \overline{\overline{R}})

The result then follows by results of the paper [5] for the intersections of plane projective curves. This shows (∗∗). We now need to prove that, for an algebraic form ϕ_{λ} in Σ and a branch γ_{p}^{j} of C;

I_{γ_{p}^{j}}(C, ϕ_{λ} ▪ G) = I_{γ_{p}^{j}}(C, ϕ_{λ}) + I_{γ_{p}^{j}}(C, G)

This follows by exactly the same argument, reducing to the case of intersections between plane projective curves and using the results of [5]. The result is then shown.

□

Theorem 3.14. Birational Invariance of a g_{n}^{r}

Let Φ : C_{1} ↭ C_{2} be a birational map between projective algebraic curves. Then, given a g_{n}^{r} on C_{2}, there exists a canonically defined g_{n}^{r} on C_{1}, depending only on the class [Φ] of the birational map. Conversely, given a g_{n}^{r} on C_{1}, there exists a canonically defined g_{n}^{r} on C_{2}, depending only on the class [Φ^{−1}] of the birational map. Moreover, these correspondences are inverse.

Proof. By Lemma 5.7 of [6], [Φ] induces a bijection;

[Φ]^{∗} : ⋃_{O∈C_{2}} γ_{O} → ⋃_{O∈C_{1}} γ_{O}

of branches, with inverse given by [Φ^{−1}]^{∗}.

Then [Φ]^{∗} extends naturally to a map on weighted sets of degree n by the formula;

[Φ]^{∗}(n_{1}γ_{p_{1}}^{j_{1}} + … + n_{r}γ_{p_{r}}^{j_{r}}) = n_{1}[Φ]^{∗}(γ_{p_{1}}^{j_{1}}) + … + n_{r}[Φ]^{∗}(γ_{p_{r}}^{j_{r}})

for a linear combination of branches {γ_{p_{1}}^{j_{1}}, …, γ_{p_{r}}^{j_{r}}} with n = n_{1} + … + n_{r}. Therefore, given a g_{n}^{r} on C_{2}, we obtain a canonically defined collection [Φ]^{∗}(g_{n}^{r}) of weighted sets on C_{1} of degree n (∗). It is trivial to see that [Φ^{−1}]^{∗} ∘ [Φ]^{∗}(g_{n}^{r}) recovers the original g_{n}^{r} on C_{2}, by the fact the map [Φ]^{∗} on branches is invertible, with inverse given by [Φ^{−1}]^{∗}. Let C^{ns} be a non-singular model of C_{1} and C_{2} with morphisms Φ_{1} : C^{ns} → C_{1} and Φ_{2} : C^{ns} → C_{2} such that Φ ∘ Φ_{1} = Φ_{2} and Φ^{−1} ∘ Φ_{2} = Φ_{1} as birational maps (see the proof of Lemma 5.7 in [6]). We then have that [Φ]^{∗}(g_{n}^{r}) = [Φ_{1}^{−1}]^{∗} ∘ [Φ_{2}]^{∗}(g_{n}^{r}). It remains to prove that this collection given by (∗) defines a g_{n}^{r} on C_{1}. We will prove first that [Φ_{2}]^{∗}(g_{n}^{r}) defines a g_{n}^{r} on C^{ns} (†). Let the original g_{n}^{r} on C_{2} be defined by a linear system Σ, having finite intersection with C_{2}, such that dimension(Σ) = r and degree(Σ) = n′, after removing some fixed branch contribution of multiplicity n′ − n. We may assume that n′ = n, as if the fixed branch contribution in question is given by W_{0} and g_{n}^{r} ∪ W_{0} = g_{n′}^{r}, then [Φ_{2}]^{∗}(g_{n}^{r}) ∪ [Φ_{2}]^{∗}(W_{0}) = [Φ_{2}]^{∗}(g_{n′}^{r}), hence it is sufficient to prove that [Φ_{2}]^{∗}(g_{n′}^{r}) defines a g_{n′}^{r}. Let W_{1} be the fixed branch contribution of the g_{n}^{r} on C_{2} and let g_{n′′}^{r} ⊂ g_{n}^{r} be obtained by removing this fixed branch contribution. It will be sufficient to prove that [Φ_{2}]^{∗}(g_{n′′}^{r}) defines a g_{n′′}^{r} on C^{ns} as [Φ_{2}]^{∗}(g_{n}^{r}) = [Φ_{2}]^{∗}(g_{n′′}^{r})∪[Φ_{2}]^{∗}(W_{1}) and we may then use Lemma 3.13. Let Φ_{Σ_{1}} and Φ_{Σ_{2}} be presentations of the morphisms Φ_{1} and Φ_{2}. We may assume that Base(Σ_{1}) and Base(Σ_{2}) are disjoint. Let \{\overline{ϕ_{λ}}\} denote the lifted family of forms on C^{ns}, defined by the linear system Σ and the presentation Φ_{Σ_{2}}. We claim that [Φ_{2}]^{∗}(g_{n′′}^{r}) is defined by this system after removing its fixed branch contribution. In order to see this, we first show that for any branch γ_{p}^{j} of C;

I_{γ_{p}^{j}}^{Σ,mobile}(C, ϕ_{λ}) = I_{p_{j}}^{Σ,mobile}(C^{ns}, \overline{ϕ_{λ}}) (∗) (1)

where p_{j} corresponds to γ_{p}^{j} in the fibre Γ_{[Φ_{2}]}(x, p), see Section 5 of [6]. By Definition 2.20 and Lemma 5.23 of [6], we have that;

I_{p_{j}}^{Σ,mobile}(C^{ns}, \overline{ϕ_{λ}}) = Card(C^{ns} ∩ (V_{p_{j}} \ p_{j}) ∩ \overline{ϕ_{λ′}}) for λ′ ∈ V_{λ}, generic in P ar_{Σ}

I_{γ_{p}^{j}}^{Σ,mobile}(C, ϕ_{λ}) = Card(C ∩(γ_{p}^{j} \p)∩ϕ_{λ′}) for λ′ ∈ V_{λ}, generic in P ar_{Σ}

As (γ_{p}^{j} \ p) is in biunivocal correspondence with (V_{p_{j}} \ p_{j}) under the morphism Φ_{2}, we obtain immediately the result (∗). Now, using Lemma 5.27 of [6], we have that, if γ_{p}^{j} appears in a weighted set W_{λ} of the g_{n′′}^{r} with multiplicity s, then the corresponding branch γ_{p_{j}} appears in the weighted set [Φ_{2}]^{∗}(W_{λ}) with multiplicity equal to s = I_{p_{j}}^{mobile}(C^{ns}, \overline{ϕ_{λ}}). Again, using Lemma 5.27 of [6], we obtain that [Φ_{2}]^{∗}(W_{λ}) is given by C^{ns} ⊓ \overline{ϕ_{λ}}, after removing all fixed point contributions of the linear system Σ. We, therefore, obtain that [Φ_{2}]^{∗}(g_{n′′}^{r}) is defined by Σ, after removing all fixed branch contributions, as required. This proves (†). We now claim that, for the given g_{n}^{r} on C^{ns}, [Φ_{1}^{−1}]^{∗}(g_{n}^{r}) defines a g_{n}^{r} on C_{1}, (††). Let Φ_{Σ_{3}} be a presentation of the morphism Φ_{1}^{−1}. If ϕ_{λ} is a form belonging to the linear system Σ defined on C^{ns}, using the presentations Φ_{Σ_{1}} and Φ_{Σ_{3}} of Φ_{1} and Φ_{1}^{−1}, we obtain a lifted form \overline{ϕ_{λ}} on C_{1} and a lifted form \overline{\overline{ϕ_{λ}}} on C^{ns} again. We now claim that, for p ∈ C^{ns};

I_{p}^{Σ,mobile}(C^{ns}, ϕ_{λ}) = I_{p}^{Σ,mobile}(C^{ns}, \overline{\overline{ϕ_{λ}}}) (2)

In order to see this, first observe that we can obtain the lifted system of forms \{\overline{\overline{ϕ_{λ}}}\} directly from the linear system Σ_{4}, obtained by composing bases of the linear systems Σ_{1} and Σ_{3}. The corresponding morphism Φ_{Σ_{4}} defines a birational map of C^{ns} to itself, which is equivalent to the identity map Id. Now the result follows immediately from Definition 2.20 and Lemma 2.16 of [6], both multiplicities are witnessed inside the canonical set W of Φ_{Σ_{4}}, which, in this case, is just the domain of definition of Φ_{Σ_{4}} on C^{ns}, see Definition 1.30 of [6]. Now, returning to the proof of (††), we may suppose that the given g_{n}^{r} on C^{ns} is defined by the linear system Σ, after removing all fixed branch contributions.

Combining (1) and (2), we have that, for a branch γ_{p}^{j} of C_{1};

I_{γ_{p}^{j}}^{Σ,mobile}(C_{1}, \overline{ϕ_{λ}}) = I_{p_{j}}^{Σ,mobile}(C^{ns}, \overline{\overline{ϕ_{λ}}}) = I_{p_{j}}^{Σ,mobile}(C^{ns}, ϕ_{λ})

The result now follows from the same argument as above, using Lemma 5.27 of [6]. This completes the theorem.

Remarks 3.15. Using the quoted Theorem 1.33 of [6], one can use the Theorem to reduce calculations involving g_{n}^{r} on projective algebraic curves to calculations on plane projective curves. This idea is central to the philosophy of the ”Italian School” of algebraic geometry.

□

We finally note the following;

Lemma 3.16. For a given g_{n}^{r}, we always have that r ≤ n.

Proof. The proof is almost identical to Lemma 2.24 of [6]. We leave the details to the reader.

□

4. A Theory of Complete Linear Series on an Algebraic Curve

We now develop further the theory of g_{n}^{r} on an algebraic curve C, analogously to classical results for divisors on non-singular algebraic curves. We will first assume that C is a plane projective algebraic curve, defined by some homogeneous polynomial F (X, Y, Z). Without loss of generality, we will use the coordinates x = X/Z and y = Y/Z for local calculations on the curve C, defined in this system by f(x, y) = 0. Using Theorem 3.14, we will later derive general results for g_{n}^{r} on an algebraic curve from the corresponding calculations for the plane case.

We consider first the case when r = 1. By results of the previous section, a g_{n}^{1} is defined by a pencil Σ of algebraic curves {ϕ(x, y) + λϕ′(x, y) = 0}_{λ∈P^{1}} (in affine coordinates), after removing some fixed branch contribution, where, by convention, we interpret the algebraic curve ϕ(x, y) + ∞ϕ′(x, y) = 0 to be ϕ′(x, y) = 0. We assume that the g_{n}^{1} is, in fact, cut out by this pencil. Now suppose that γ_{p} is a branch of C. We may assume that p corresponds to the origin O of the affine coordinate system (x, y), (use a linear transformation and the result of Lemma 4.1) By Theorem 6.1 of [6], we can find algebraic power series {x(t), y(t)}, with x(t) = y(t) = 0, parametrising γ_{p}. We can now substitute the power series in order to obtain a formal expression of the form;

\frac{ϕ(x(t),y(t))}{ϕ′(x(t),y(t))} = \frac{t^{i}u(t)}{t^{j}v(t)} = t^{i−j}u(t)v(t)^{−1}, where \{u(t), v(t), u(t)v(t)^{−1}\}

are units in L[[t]].

We then define;

(i). ord_{γ_{p}}(\frac{ϕ}{ϕ′}) = i − j,

val_{γ_{p}}(\frac{ϕ}{ϕ′}) = 0, if i > j, (\frac{ϕ}{ϕ′} has a zero of order i − j)

(ii). ord_{γ_{p}}(\frac{ϕ}{ϕ′}) = j − i,

val_{γ_{p}}(\frac{ϕ}{ϕ′}) = ∞, if i < j, (\frac{ϕ}{ϕ′} has a pole of order j −i)

(iii). ord_{γ_{p}}(\frac{ϕ}{ϕ′}) = ord_{t}(h(t) − h(0)),

val_{γ_{p}}(\frac{ϕ}{ϕ′}) = h(0), if i = j and h(t) = u(t)v(t)^{−1}

Observe that in all cases, ord_{γ_{p}} gives a positive integer, while val_{γ_{p}} determines an element of P^{1}. In order to see that this construction does not depend on the particular power series representation of the branch, we require the following lemma;

Lemma 4.1. Let {C, γ_{p}, ϕ, ϕ′, g_{n}^{1}, Σ} be as defined above, then;

ord_{γ_{p}}(\frac{ϕ}{ϕ′}) = I_{γ_{p}}(C, ϕ − λϕ′), if γ_{p} is not a base branch for the g_{n}^{1} and \frac{ϕ}{ϕ′} (p) = val_{γ_{p}}(\frac{ϕ}{ϕ′}) = λ.

ord_{γ_{p}}(\frac{ϕ}{ϕ′}) = I_{γ_{p}}^{Σ,mobile}(C, ϕ−λϕ′), if γ_{p} is a base branch for the g_{n}^{1} and λ = val_{γ_{p}}(\frac{ϕ}{ϕ′}) is unique such that,

for μ ≠ λ;

I_{γ_{p}}(C, ϕ − λϕ′) > I_{γ_{p}}(C, ϕ − μϕ′).

Proof. Suppose that γ_{p} is not a base branch for the g_{n}^{1}, then \frac{ϕ}{ϕ′} (p) = λ is well defined, if we interpret (c/0) = ∞ for c ≠ 0, and ϕ − λϕ′ is the unique curve in the pencil passing through p. It is trivial to check, using the facts that ϕ(p) = ϕ(x(0), y(0)) and ϕ′(p) = ϕ′(x(0), y(0)), that, in all cases, val_{γ_{p}}(\frac{ϕ}{ϕ′}) = λ as well. By Theorem 6.1 of [6], we have

28 TRISTRAM DE PIRO

that; I_{γ_{p}}(C, ϕ − λϕ′) = ord_{t}[(ϕ − λϕ′)(x(t), y(t))] If λ = 0, then ϕ(p) = 0 and ϕ′(p) ≠ 0, hence, by a straightfor-

ward algebraic calculation, ϕ(x(t), y(t)) = t^{i}u(t), for some i ≥ 1, and ϕ′(x(t), y(t)) = v(t) for {u(t), v(t)} units in L[[t]]. Therefore, ord_{γ_{p}}(\frac{ϕ}{ϕ′}) = ord_{t}ϕ(x(t), y(t)) and the result follows.

If λ = ∞, then ϕ(p) ≠ 0 and ϕ(p) = 0, hence, ϕ(x(t), y(t)) = u(t) and ϕ′(x(t), y(t)) = t^{j}v(t), for some j ≥ 1, and {u(t), v(t)} units in L[[t]]. Therefore, ord_{γ_{p}}(\frac{ϕ}{ϕ′}) = ord_{t}ϕ′(x(t), y(t)) and the result follows.

If λ ≠ {0, ∞}, then ϕ(x(t), y(t)) = u(t) and ϕ′(x(t), y(t)) = v(t) with {u(t), v(t)} units in L[[t]]. As v(t) is a unit in L[[t]], we have that;

ord_{t}(\frac{u(t)}{v(t)} − \frac{u(0)}{v(0)}) = ord_{t}(v(t)(\frac{u(t)}{v(t)} − \frac{u(0)}{v(0)})) = ord_{t}(u(t) − \frac{u(0)}{v(0)}v(t))

Hence, by definition of ord_{γ_{p}};

ord_{γ_{p}}(\frac{ϕ}{ϕ′}) = ord_{t}[(ϕ − λϕ′)(x(t), y(t))]

and the result follows.

Now suppose that γ_{p} is a base branch for the g_{n}^{1}, then ϕ(p) = ϕ′(p) = 0 and we have that ϕ(x(t), y(t)) = t^{i}u(t) and ϕ′(x(t), y(t)) = t^{j}v(t), for some i, j ≥ 1 and {u(t), v(t)} units in L[[t]]. Again, we divide the proof into the following cases;

i > j. In this case, by definition, val_{γ_{p}}(\frac{ϕ}{ϕ′}) = 0. We compute; ord_{t}(ϕ(x(t), y(t)) − λϕ′(x(t), y(t))) = ord_{t}(t^{i}u(t) − λt^{j}v(t)) When λ = 0, we obtain, by Theorem 6.1 of [6], that I_{γ_{p}}(C, ϕ) = i

and, for λ ≠ 0, that I_{γ_{p}}(C, ϕ − λϕ′) = j. Using Lemma 5.27 of [6], we obtain that I_{γ_{p}}^{Σ,mobile}(C, ϕ) = i − j = ord_{γ_{p}}(\frac{ϕ}{ϕ′}), as required.

i < j. In this case, by definition, val_{γ_{p}}(\frac{ϕ}{ϕ′}) = ∞. The computation for ord_{γ_{p}} is similar, with the critical value being λ = ∞. i = j. We compute;

ord_{t}(ϕ(x(t), y(t)) − λϕ′(x(t), y(t))) = ord_{t}[t^{i}(u(t) − λv(t))]

Again, there exists a unique value of λ = \frac{u(0)}{v(0)} = val_{γ_{p}}(\frac{ϕ}{ϕ′}) ≠ \{0, ∞\} such that ord_{t}(u(t)−λv(t)) = k ≥ 1. By the same calculation as above, we have that I_{γ_{p}}^{Σ,mobile}(C, ϕ − λϕ′) = k, for this critical value of λ. By a similar algebraic calculation to the above, using the fact that v(t) is a unit, we also compute ord_{γ_{p}}(\frac{ϕ}{ϕ′}) = k, hence the result follows.

□

We now show the following;

Lemma 4.2. Given any algebraic curve C ⊂ P^{w}, with function field L(C), for a non-constant rational function f ∈ L(C) and a branch γ_{p}, we can unambiguously define ord_{γ_{p}}(f) and val_{γ_{p}}(f).

Proof. The proof is similar to the above. We may, without loss of generality, assume that p corresponds to the origin of a coordinate system (x_{1}, …, x_{w}). Using Theorem 6.1 of [6], we can find algebraic power series (x_{1}(t), …, x_{w}(t)) parametrising the branch γ_{p}. By the assumption that f is non-constant, we can find a representation of f as a rational function \frac{ϕ(x_{1},...,x_{w})}{ϕ′(x_{1},...,x_{w})} in this coordinate system, such that the pencil Σ defined by {ϕ, ϕ′} has finite intersection with C, hence defines a g_{n}^{1}. Using the method above, we can define ord_{γ_{p}}(\frac{ϕ}{ϕ′}) and val_{γ_{p}}(\frac{ϕ}{ϕ′}) for this representation. The proof of Lemma 4.1 shows that these are defined independently of the particular power series parametrising the branch. We need to check that they are also defined independently of the particular representation of f. Suppose that {ϕ_{1}, ϕ_{2}, ϕ_{3}, ϕ_{4}} are algebraic forms with the property that \frac{ϕ_{1}}{ϕ_{2}} = \frac{ϕ_{3}}{ϕ_{4}} as rational functions on C. We claim that, for any branch γ_{p} of C, ord_{γ_{p}}(\frac{ϕ_{1}}{ϕ_{2}} ) = ord_{γ_{p}}(\frac{ϕ_{3}}{ϕ_{4}} ) and val_{γ_{p}}(\frac{ϕ_{1}}{ϕ_{2}} ) = val_{γ_{p}}(\frac{ϕ_{3}}{ϕ_{4}} ), (∗). In order to see this, let U ⊂ NonSing(C) be an open subset of C, on which \frac{ϕ_{1}}{ϕ_{2}} and \frac{ϕ_{3}}{ϕ_{4}} are defined and equal. Let g_{n}^{1} and g_{m}^{1} on C be defined by the pencils Σ_{1} = {ϕ_{1} − λϕ_{2}}_{λ∈P^{1}} and Σ_{2} = {ϕ_{3} −λϕ_{4}}_{λ∈P^{1}}. Let V = U \Base(Σ_{1})∪Base(Σ_{2}). Then V ⊂ U is also an open subset of C, which we will refer to as the canonical set. Now, suppose that γ_{p} ⊂ V. We will prove (∗) for this branch. As both \frac{ϕ_{1}}{ϕ_{2}} and \frac{ϕ_{3}}{ϕ_{4}} are defined and equal at p, using the argument in Lemma 4.1, we have that val_{γ_{p}}(\frac{ϕ_{1}}{ϕ_{2}} ) = val_{γ_{p}}(\frac{ϕ_{3}}{ϕ_{4}} ). It is therefore sufficient, again by Lemma 4.1, to show that;

I_{γ_{p}}(C, ϕ_{1} − λϕ_{2}) = I_{γ_{p}}(C, ϕ_{3} − λϕ_{4}), for \frac{ϕ_{1}}{ϕ_{2}} (p) = \frac{ϕ_{3}}{ϕ_{4}} (p) = λ (†)

Suppose that I_{γ_{p}}(ϕ_{1} − λϕ_{2}) = m, then, by Lemma 5.25 of [6], we can find λ′ ∈ V_{λ} ∩ P^{1} and {p_{1}, …, p_{m}} = V ∩ V_{p} ∩ (ϕ_{1} − λ′ϕ_{2}) = 0 witnessing this multiplicity. As {p, p_{1}, …, p_{m}} lie inside V, we also have that {p_{1}, …, p_{m}} ⊂ V ∩ V_{p} ∩ (ϕ_{3} − λ′ϕ_{4}) = 0, hence I_{γ_{p}}(C, ϕ_{3} − λϕ_{4}) ≥ m. The result (†) then follows from the converse argument.

Now, suppose that γ_{p} is one of the finitely many branches of C, not lying inside V. We will just consider the case when γ_{p} is a base branch for both the g_{n}^{1} and the g_{m}^{1} defined above, the other cases being similar. In order to prove (∗) for this branch, it is sufficient, by Lemma 4.1, to show that;

I_{γ_{p}}^{Σ_{1},mobile}(C, ϕ_{1} −λϕ_{2}) = I_{γ_{p}}^{Σ_{2},mobile}(C, ϕ_{3} −μϕ_{4}), for the critical values {λ, μ}

and that the critical values {λ, μ} coincide, (††).

Using the argument to prove (†), witnessing the corresponding multiplicities in the canonical set V, it follows that for any ν ∈ P^{1};

I_{γ_{p}}^{Σ_{1},mobile}(C, ϕ_{1} − νϕ_{2}) = I_{γ_{p}}^{Σ_{2},mobile}(C, ϕ_{3} − νϕ_{4}), (†††)

If the critical values {λ, μ} were distinct, we would have that;

I_{γ_{p}}^{Σ_{1},mobile}(C, ϕ_{1} − λϕ_{2}) > I_{γ_{p}}^{Σ_{1},mobile}(C, ϕ_{1} − μϕ_{2})

|| ||

I_{γ_{p}}^{Σ_{2},mobile}(C, ϕ_{3} − λϕ_{4}) < I_{γ_{p}}^{Σ_{2},mobile}(C, ϕ_{3} − μϕ_{4})

which is clearly a contradiction. Hence, λ = μ and the result (††) follows from (†††). The lemma is shown. □

Lemma 4.3. Birational Invariance of ord_{γ_{p}} and val_{γ_{p}}

Let Φ : C_{1} ↭ C_{2} be a birational map between projective algebraic curves with corresponding isomorphisms Φ^{∗} : L(C_{2}) → L(C_{1}) and [Φ]^{∗} : ⋃_{p∈C_{2}} γ_{p} → ⋃_{q∈C_{1}} γ_{q} . Then, for non-constant f ∈ L(C_{2}) and γ_{p} a branch of C_{2}, ord_{γ_{p}}(f) = ord_{[Φ]^{∗}γ_{p}}(Φ^{∗}f) and val_{γ_{p}}(f) = val_{[Φ]^{∗}γ_{p}}(Φ^{∗}f).

A THEORY OF DIVISORS FOR ALGEBRAIC CURVES 31

Proof. Let f be represented as a rational function by \frac{ϕ_{1}}{ϕ_{2}} , as in Lemma 4.2, and consider the g_{n}^{1} on C_{2}, defined by the linear system Σ = {ϕ_{1} − λϕ_{2}}_{λ∈P^{1}}. Let Φ_{Σ_{1}} be a presentation of the birational map Φ. Using this presentation, we may lift the system Σ to a corresponding linear system \{\overline{ϕ_{1}} − λ\overline{ϕ_{2}}\}_{λ∈P^{1}}. It is trivial to check that Φ^{∗}f is represented by the rational function \frac{\overline{ϕ_{1}}}{\overline{ϕ_{2}}} . The proof of Theorem 3.14 shows that, for a branch γ_{p} of C_{2};

I_{γ_{p}}^{Σ,mobile}(C_{2}, ϕ_{1} − λϕ_{2}) = I_{[Φ]^{∗}γ_{p}}^{Σ,mobile}(C_{1}, \overline{ϕ_{1}} − λ\overline{ϕ_{2}}), (∗)

We now need to consider the following cases;

Case 1. γ_{p} and [Φ]^{∗}γ_{p} are not base branches for Σ on C_{2} and C_{1}.

Case 2. γ_{p} is not a base branch, but [Φ]^{∗}γ_{p} is a base branch for Σ on

C_{2} and C_{1}. Case 3. γ_{p} is a base branch and [Φ]^{∗}γ_{p} is a base branch for Σ on C_{2}

and C_{1}. For Case 1, we have, by Lemma 4.1 and (∗); ord_{γ_{p}}(\frac{ϕ_{1}}{ϕ_{2}} ) = I_{γ_{p}}(C_{2}, ϕ_{1} − λϕ_{2}) = I_{[Φ]^{∗}γ_{p}}(C_{1}, \overline{ϕ_{1}} − λ\overline{ϕ_{2}}) = ord_{[Φ]^{∗}γ_{p}}(\frac{\overline{ϕ_{1}}}{\overline{ϕ_{2}}} ) where \frac{ϕ_{1}}{ϕ_{2}} (p) = \frac{\overline{ϕ_{1}}}{\overline{ϕ_{2}}} (q) = val_{γ_{p}}(\frac{ϕ_{1}}{ϕ_{2}} ) = val_{γ_{q}}(\frac{\overline{ϕ_{1}}}{\overline{ϕ_{2}}} ) = λ and [Φ]^{∗}γ_{p} = γ_{q}. For Case 3, we have, by Lemma 4.1, (∗) and a similar argument to

the previous lemma, in order to show the critical value λ = val_{γ_{p}}(\frac{ϕ_{1}}{ϕ_{2}} ) is also the critical value val_{γ_{q}}(\frac{\overline{ϕ_{1}}}{\overline{ϕ_{2}}} ) for the lifted system at the corresponding branch [Φ]^{∗}γ_{p}, that;

ord_{γ_{p}}(\frac{ϕ_{1}}{ϕ_{2}} ) = I_{γ_{p}}^{Σ,mobile}(C_{2}, ϕ_{1}−λϕ_{2}) = I_{[Φ]^{∗}γ_{p}}^{Σ,mobile}(C_{1}, \overline{ϕ_{1}}−λ\overline{ϕ_{2}}) = ord_{[Φ]^{∗}γ_{p}}(\frac{\overline{ϕ_{1}}}{\overline{ϕ_{2}}} ) Case 2 is similar, we leave the details to the reader. The lemma now follows from the previous lemma, that the definitions

of ord_{γ_{p}}(f), ord_{[Φ]^{∗}γ_{p}}(Φ^{∗}f),val_{γ_{p}}(f) and val_{[Φ]^{∗}γ_{p}}(Φ^{∗}f) are independent of their particular representations.

□

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We now show; Lemma 4.4. flatness

Let C be a projective algebraic curve, then, to any non-constant rational function f on C, we can associate a g_{n}^{1} on C, which we will denote by (f), where n = deg(f). Proof. We define the weighted set (f = λ) as follows;

(f = λ) := {n_{γ_{1}}, …, n_{γ_{r}}}

where {γ_{1}, …, γ_{r}} = {γ : val_{γ}(f) = λ} and n_{γ} = ord_{γ}(f).

As λ varies over P^{1}, we obtain a series of weighted sets W_{λ} on C. We claim that this series does in fact define a g_{n}^{1}. In order to see this, let f be represented as a rational function by \frac{ϕ}{ϕ′}. As before, we consider the pencil Σ of forms defined by (ϕ − λϕ′)_{λ∈P^{1}}. We claim that the series is defined by this system Σ, after removing its fixed branch contribution, (∗). In order to see this, we compare the weighted sets (f = λ) and C ⊓(ϕ−λϕ′). For a branch γ_{p} which is not a fixed branch of the system Σ, we have, using Lemmas 4.1 and 4.2, that;

γ_{p} ∈ (f = λ) iff val_{γ_{p}}(f) = λ iff \frac{ϕ}{ϕ′} (p) = λ iff p ∈ C ∩ (ϕ − λϕ′)

In this case, by Lemmas 4.1 and 4.2, we have that;

n_{γ_{p}} = ord_{γ_{p}}(f) = ord_{γ_{p}}(\frac{ϕ}{ϕ′}) = I_{γ_{p}}(C, ϕ − λϕ′)

For a branch γ_{p} which is a fixed branch of the system Σ, we have, by Lemmas 4.1 and 4.2, that;

γ_{p} ∈ (f = λ) iff val_{γ_{p}}(\frac{ϕ}{ϕ′}) = λ iff p ∈ C ∩ (ϕ − λϕ′) and λ is a critical value for the system Σ at γ_{p}.

In this case, by Lemmas 4.1 and 4.2, we have that;

n_{γ_{p}} = ord_{γ_{p}}(f) = ord_{γ_{p}}(\frac{ϕ}{ϕ′}) = I_{γ_{p}}^{Σ,mobile}(C, ϕ − λϕ′) (1)

Let I_{γ_{p}} = min_{μ∈P^{1}}I_{γ_{p}}(C, ϕ − μϕ′) be the fixed branch contribution of Σ at γ_{p}. Then, at the critical value λ for the system Σ; I_{γ_{p}}^{Σ,mobile}(ϕ − λϕ′) = I_{γ_{p}}(C, ϕ − λϕ′) − I_{γ_{p}} (2)

Hence, the result (∗) follows from (1), (2) and the definition of C ⊓ (ϕ − λϕ′).

Finally, we show that n = deg(f). Let Γ_{f} be the correspondence determined by the rational map f : C ⇝ P^{1}. By classical arguments, deg(f) is equal to the cardinality of the generic fibre Γ_{f}(λ), for λ ∈ P^{1}. Fixing a presentation \frac{ϕ}{ϕ′} for f, if U ⊂ NonSing(C) is the canonical set for this presentation, one may assume that the generic fibre Γ_{f}(λ) lies inside U. By Lemma 2.17 of [6], one may also assume that the corresponding weighted set of the g_{n}^{1} defined by (f = λ) consists of n distinct branches, centred at the points of the generic fibre Γ_{f}(λ). Therefore, the result follows.

□

Remarks 4.5. By convention, for a non-zero rational function c ∈ L \ {0}, we define (c = 0) and (c = ∞) to be the empty weighted sets. The notion of a weighted set in a g_{n}^{1}, generalises the classical notion of the divisor on a non-singular curve. Using the above theorem, we can make sense of the notion of linear equivalence of weighted sets.

We make the following definition;

Definition 4.6. Linear equivalence of weighted sets

Let C be an algebraic curve and let A and B be weighted sets on C of the same total multiplicity. We define A ≡ B if there exists a g_{n}^{r} on C such that A and B belong to this g_{n}^{r} as weighted sets.

Theorem 4.7. Let hypotheses be as in the previous definition. If A ≡ B, then there exists a rational function g on C, such that A is defined by (g = 0) and B is defined by (g = ∞), possibly after adding some fixed branch contribution.

Proof. If r = 0 in the definition, then we must have that A = B. Hence, we obtain the statement of the theorem by adding the fixed branch contribution A to the empty g_{0}^{0}, defined by (c = 0) = (c = ∞), for a non-constant c ∈ L^{∗}. Otherwise, by the definition of a g_{n}^{r}, we may, without loss of generality, find a pencil Σ of algebraic forms, {ϕ − λϕ′}_{λ∈P^{1}}, having finite intersection with C, such that;

A = C ⊓ (ϕ − λ_{1}ϕ), B = C ⊓ (ϕ − λ_{2}ϕ′) (λ_{1} ≠ λ_{2})

Let f be the rational function on C defined by \frac{ϕ}{ϕ′}. If A and B have no branches in common (with multiplicity), (†), then the pencil Σ can have no fixed branches and, by Lemma 4.4, we have that;

A = (f = λ_{1})

B = (f = λ_{2}) (λ_{1} ≠ λ_{2})

Now we can find an algebraic automorphism α of P^{1}, taking λ_{1} to 0 and λ_{2} to ∞. We will assume that {λ_{1}, λ_{2}} ≠ ∞, in which case α can be given, for a coordinate z on P^{1}, by the Mobius transformation \frac{z−λ_{1}}{z−λ_{2}} . The other cases are left to the reader. Let g be the rational function on C defined by α ∘ f. Now, suppose that γ is a branch of C, with val_{γ}(f) = λ and ord_{γ}(f) = m. Then, we claim that val_{γ}(g) = α(λ) and ord_{γ}(g) = m, (∗). If λ ≠ {λ_{2}, ∞}, using the method before Lemma 4.1, we obtain the following power series representation of g at γ;

\frac{(λ+μt^{m}+o(t^{m}))−λ_{1}}{(λ+μt^{m}+o(t^{m}))−λ_{2}} = [(λ−λ_{1})+μt^{m}+o(t^{m})]▪ \frac{1}{(λ−λ_{2})}[1− \frac{μ}{(λ−λ_{2})}t^{m}+o(t^{m})]

= \frac{λ−λ_{1}}{λ−λ_{2}} + t^{m}[\frac{μ(λ−λ_{2})−μ(λ−λ_{1})}{(λ−λ_{2})^{2}}] + o(t^{m})
= \frac{λ−λ_{1}}{λ−λ_{2}} + t^{m}[\frac{μ(λ_{1}−λ_{2})}{(λ−λ_{2})^{2}}] + o(t^{m})

and the claim (∗) follows from the assumption that λ_{1} ≠ λ_{2}. If λ = λ_{2}, we obtain the following power series representation of g at γ;

\frac{(λ+μt^{m}+o(t^{m}))−λ_{1}}{(μt^{m}+o(t^{m}))} = \frac{1}{t^{m}} ▪ [(λ − λ_{1}) + μt^{m} + o(t^{m})] ▪ [μ + o(1)]^{−1}

which gives that val_{γ}(g) = ∞ = α(λ_{2}) and ord_{γ}(g) = m, using the fact that λ ≠ λ_{1}. Finally, if λ = ∞, the Mobius transformation at ∞ is given by \frac{\frac{1}{z} −λ_{1}}{\frac{1}{z} −λ_{2}} = \frac{1−λ_{1}z}{1−λ_{2}z} and g may be represented at γ by \frac{ϕ−λ_{1}ϕ′}{ϕ−λ_{2}ϕ′}. We then obtain the power series representation of g at γ;

\frac{(t^{i}u(t)−λ_{1}t^{i+m}v(t))}{(t^{i}u(t)−λ_{2}t^{i+m}v(t))} = \frac{(u(t)−λ_{1}t^{m}v(t))}{(u(t)−λ_{2}t^{m}v(t))} = \frac{[1−λ_{1}t^{m} \frac{v(t)}{u(t)} ]}{[1−λ_{2}t^{m} \frac{v(t)}{u(t)} ]}

= 1 + (λ_{2} − λ_{1})t^{m}w(t) + o(t^{m}), for {u(t), v(t), w(t)}

units in L[[t]]

which gives that val_{γ}(g) = 1 = α(∞) and ord_{γ}(g) = m, using the fact that λ_{1} ≠ λ_{2} again. This gives the claim (∗). It follows that the weighted sets (f = λ) correspond exactly to the weighted sets (g = α(λ)), in particularly the g_{n}^{1} defined by (f) and (g), as in Lemma 4.4, is the same. With this new parametrisation of the g_{n}^{1}, we then have that;

A = (g = 0)

B = (g = ∞)

Hence, the result follows, with the assumption (†). If A and B have branches in common, with multiplicity, we let A ∩ B denote the weighted set consisting of these common branches (with multiplicity). Then, the same argument holds, replacing A by A \ B = A − (A ∩ B) and B by B \ A = B − (A ∩ B). After adding the fixed branch contribution (A ∩ B) to the g_{n}^{1} defined by (g), we then obtain the result. Note that, by Lemma 3.13, this addition defines a g_{n+n′}^{1}, where n′ is the total multiplicity of (A ∩ B).

□

Remarks 4.8. The definition we have given of linear equivalence of weighted sets on a projective algebraic curve C generalises the modern definition of linear equivalence for effective divisors on a smooth projective algebraic curve. More precisely we have;

Modern Definition; Let A and B be effective divisors on a smooth projective algebraic curve C, then A ≡ B iff A − B = div(g), for some g ∈ L(C)^{∗}.

See, for example, p161 of [10] for relevant definitions and notation. We now show that our definition is the same in this case. First, observe that there exists a natural bijection between the set of effective divisors on C, in the sense of [10], and the collection of weighted sets on C, (∗). This follows immediately from the fact, given in Lemma 5.29 of [6], that, for each point p ∈ C, there exists a unique branch γ_{p}, centred at p. Secondly, observe that the notion of div(g), for g ∈ L(C), as given in [10], is the same as the notion of div(g) which we give in Definition 4.9 below, (taking into account the identification (∗)), (†). This amounts to checking that, for a point p ∈ C, with corresponding branch γ_{p}; v_{p}(g) = ord_{γ_{p}}(g) (††)

where v_{p}(g) is defined in p152 of [10]. First, one can use the fact, given in Lemma 4.9 of [6], together with remarks from the final section of this paper, that there exists a birational map ϕ : C ↭ C′, such that C′ is a plane projective algebraic curve, and p corresponds to a nonsingular point p′ ∈ C′ with {p, p′} lying inside the canonical sets associated to ϕ. Using the calculation given below, in Lemma 4.10, for ord_{γ_{p}}, and the definition of v_{p}, one can assume that v_{p}(g) ≥ 0 and g ∈ O_{p,C}. Let g′ ∈ L(C′) denote the corresponding rational function to g on L(C). It is then a trivial algebraic calculation, using the fact that the local rings O_{p,C} and O_{p′,C′} are isomorphic, to show that v_{p}(g) = v_{p′}(g′). It also follows from Lemma 4.3 that ord_{γ_{p}}(g) = ord_{γ_{p′}}(g′). Hence, it is sufficient to check (††) for the plane projective curve C′. We may, without loss of generality, assume that v_{p′}(g′) ≥ 1 and that g′ is represented in some choice of affine coordinates {x, y} by the polynomial q(x, y). If Q(X, Y, Z) denotes the projective equation of this polynomial and p′ corresponds to the origin of this coordinate system, then;

v_{p′}(g′) = I_{p′}(C, Q) = length(\frac{L[x,y]}{<h,q>})

where h is a defining equation for C′ in the coordinate system {x, y} and I_{p′} is the algebraic intersection multiplicity. It also follows from Lemma 4.1, that;

ord_{γ_{p′}}(g′) = I_{γ_{p′}}(C, Q)

Hence, it is sufficient to check that;

I_{p′}(C, Q) = I_{γ_{p′}}(C, Q)

This calculation was done in Theorem 2.10, hence (††) and therefore (†) is shown. Thirdly, it remains to check that the definitions of linear equivalence are the same. In order to see this, observe that we can write (for effective divisors or weighted sets A and B);

A−B = (A\B)+(A∩B)]−[(B\A)+(A∩B)] = (A\B)−(B\A), (†††)

If A ≡ B in the sense of weighted sets (Definition 4.6), then the calculation (†††) (which removes the fixed branch contribution) and Theorem 4.7 shows that A − B = div(g), for some rational function g ∈ L(C), where, here, div(g) is as defined in Definition 4.9. By (†), it then follows that A ≡ B as effective divisors. Conversely, if A ≡ B as effective divisors, then there exists a rational function g ∈ L(C) such that A − B = div(g), in the sense of the modern definition given above. The above calculations (†††) and (†) then show that div(g) = (A \ B) − (B \ A), in the sense of Definition 4.9 below. It follows, by Lemma 4.4, that there exists a g_{n}^{1} to which (A \ B) and (B \ A) belong as weighted sets. Adding the fixed branch contribution (A∩B) to this g_{n}^{1}, we then obtain that A ≡ B in the sense of Definition 4.6, as required.

Definition 4.9. Let C be a projective algebraic curve and let f be a non-zero rational function on C. Then we define div(f) to be the weighted set A − B where;

A = (f = 0), B = (f = ∞)

We now require the following lemma;

Lemma 4.10. Let C be a projective algebraic curve, and let f and g be non-zero rational functions on C. Then;

div(\frac{1}{f}) = −div(f)

div(fg) = div(f) + div(g)

div(\frac{f}{g}) = div(f) − div(g)

Proof. In order to prove the first claim, it is sufficient to show that, for a branch γ of C;

val_{γ}(f) = 0 iff val_{γ}(\frac{1}{f}) = ∞

val_{γ}(f) = ∞ iff val_{γ}(\frac{1}{f}) = 0

and ord_{γ} is preserved in both cases. This follows trivially from the relevant power series calculation at a branch. Namely, we can represent f by \frac{ϕ}{ϕ′} and \frac{1}{f} by \frac{ϕ′}{ϕ}. Substituting the branch parametrisation, we obtain that;

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val_{γ}(f) = 0, ord_{γ}(f) = m iff f ∼ t^{m}u(t), m ≥ 1, u(t) ∈ L[[t]] a unit. iff \frac{1}{f} ∼ t^{−m}u(t)^{−1}

iff val_{γ}(f) = ∞, ord_{γ}(f) = m

and the calculation for val_{γ}(f) = ∞, ord_{γ}(f) = m is similar. In order to prove the second claim, we need to verify the following

cases at a branch γ of C;

Case 1. If val_{γ}(f) = val_{γ}(g) ∈ {0, ∞}, ord_{γ}(f) = m and ord_{γ}(g) = n

then val_{γ}(fg) ∈ {0, ∞} and ord_{γ}(fg) = m + n Case 2. If val_{γ}(f) ≠ val_{γ}(g) ∈ {0, ∞}, ord_{γ}(f) = m and ord_{γ}(g) = n

then val_{γ}(fg) ∈ {0, ∞} and ord_{γ}(fg) = |m − n| Case 3. If exactly one of val_{γ}(f) and val_{γ}(g) is in {0, ∞}, with

ord_{γ}(f) or ord_{γ}(g) = m

then val_{γ}(fg) ∈ {0, ∞}, with ord_{γ}(fg) = m.

Case 4. If neither of val_{γ}(f) and val_{γ}(g) are in {0, ∞}

then val_{γ}(fg) is not in {0, ∞} If f is represented by \frac{ϕ}{ϕ′} and g is represented by \frac{ψ}{ψ′}, then we can rep-

resent fg by \frac{ϕψ}{ϕ′ψ′}. The proof of these cases then follow by elementary power series calculations at the branch γ. For example, for Case 2, if val_{γ}(f) = 0 and ord_{γ}(f) = m, val_{γ}(g) = ∞ and ord_{γ}(g) = n, then we have;

f ∼ t^{n}u(t), g ∼ t^{−m}v(t), fg ∼ t^{n}t^{−m}u(t)v(t) = t^{n−m}w(t),

for {u(t), v(t), w(t)} units in L[[t]].

The third claim follows from the first two claims.

□

We now claim the following; Theorem 4.11. Transitivity of Linear Equivalence

Let C′ be an algebraic curve. If A, B, C are weighted sets on C′ of the same total multiplicity, then, if A ≡ B and B ≡ C, we must have that A ≡ C.

Proof. By Theorem 4.7, we can find rational functions f and g on C′, such that;

(A \ B) − (B \ A) = div(f)

(B \ C) − (C \ B) = div(g)

By Lemma 4.10, we have that;

div(fg) = (A \ B) − (B \ A) + (B \ C) − (C \ B)

By drawing a Venn diagram, one easily checks that;

(A \ B) − (B \ A) = (A ∩ B^{c} ∩ C^{c}) + (A ∩ B^{c} ∩ C) − (A^{c} ∩ B ∩ C^{c}) −

(A^{c} ∩ B ∩ C)

+ (B \ C) − (C \ B) = (A ∩ B ∩ C^{c}) + (A^{c} ∩ B ∩ C^{c}) − (A^{c} ∩ B^{c} ∩ C) −

(A ∩ B^{c} ∩ C)

||

(A \ C) − (C \ A) = (A ∩ B^{c} ∩ C^{c}) + (A ∩ B ∩ C^{c}) − (A^{c} ∩ B^{c} ∩ C) − (A^{c} ∩ B ∩ C)

Hence, div(fg) = (A \ C) − (C \ A). Now, given the g_{n}^{1} defined by the rational function fg, as in Lemma 4.4, it follows that (A \ C) and (C \ A) belong to this g_{n}^{1} as weighted sets. We can now add the fixed branch contribution A ∩ C to this g_{n}^{1}, giving a g_{n+n′}^{1}, to which A and C belong as weighted sets. Therefore, the result follows.

□

As an immediate corollary, we have;

Theorem 4.12. Let C be a projective algebraic curve, then ≡ is an equivalence relation on weighted sets for C of a given multiplicity.

We also have;

40 TRISTRAM DE PIRO

Theorem 4.13. Linear Equivalence preserved by Addition

Let C′ be a projective algebraic curve and suppose that {A, B, C, D} are weighted sets on C′ with;

A ≡ B and C ≡ D then; A + C ≡ B + D

Proof. By Definition 4.6, we can find a g_{n}^{r} containing C and D as weighted sets. If s is the total multiplicity of A, then, by Lemma 3.13, we can add the weighted set A as a fixed branch contribution to this g_{n}^{r} and obtain a g_{n+s}^{r}, containing A + C and A + D as weighted sets. Hence, by Definition 4.6 again, we have that;

A + C ≡ A + D (1) Similarily, one shows, by adding D as a fixed branch contribution to

the g_{n′}^{r′} containing A and B as weighted sets, that; A + D ≡ B + D (2) The result then follows immediately by combining (1), (2) and using

Theorem 4.11.

□

We now develop further the theory of g_{n}^{r} on a projective algebraic curve C. We begin with the following definition;

Definition 4.14. Subordinate g_{n}^{r} Let {g_{n}^{r}, g_{n}^{t}} be given on C with the same order n. Then we say that; g_{n}^{r} ⊆ g_{n}^{t} if every weighted set in g_{n}^{r} is included in the weighted sets of the g_{n}^{t}. We now claim the following;

A THEORY OF DIVISORS FOR ALGEBRAIC CURVES 41

Theorem 4.15. Amalgamation of g_{n}^{r}

Let {g_{n}^{r}, g_{n}^{s}} be given on C, having a common weighted set G, then there exists t with r ≤ t, s ≤ t and a g_{n}^{t} such that g_{n}^{r} ⊆ g_{n}^{t} and g_{n}^{s} ⊆ g_{n}^{t}. Proof. Assume first that {g_{n}^{r}, g_{n}^{s}} have no fixed branch contribution and are defined exactly by linear systems. Then we can find algebraic forms {ϕ_{0}, ψ_{0}} such that;

G = (C ⊓ ϕ_{0} = 0) = (C ⊓ ψ_{0} = 0) and; g_{n}^{r} is defined by C ⊓ (ϵ_{0}ϕ_{0} + ϵ_{1}ϕ_{1} + … + ϵ_{r}ϕ_{r} = 0) g_{n}^{s} is defined by C ⊓ (η_{0}ψ_{0} + η_{1}ψ_{1} + … + η_{s}ψ_{s} = 0) Now consider the linear system Σ defined by; ϵϕ_{0}ψ_{0} + ψ_{0}(ϵ_{1}ϕ_{1} + … + ϵ_{r}ϕ_{r}) + ϕ_{0}(η_{1}ψ_{1} + … + η_{s}ψ_{s}) = 0 and let g_{m}^{t} be defined by Σ. As deg(ψ_{0}ϕ_{0}) = deg(ψ_{0}) + deg(ϕ_{0}), we

have that m = 2n. We claim that the fixed branch contribution of g_{2n}^{t} is exactly G, (∗). In order to see this, observe that we can write an algebraic form in Σ as;

ψ_{0}ϕϵ̄ + ϕ_{0}ψη̄ If γ is a branch counted w-times in G, then, using the proof at the

end of Lemma 3.13 and linearity of multiplicity at a branch, see [6]; I_{γ}(C, ψ_{0}ϕϵ̄) = I_{γ}(C, ψ_{0}) + I_{γ}(C, ϕϵ̄) ≥ w I_{γ}(C, ϕ_{0}ψη̄) = I_{γ}(C, ϕ_{0}) + I_{γ}(C, ψη̄) ≥ w I_{γ}(C, ψ_{0}ϕϵ̄ + ϕ_{0}ψη̄) = min{I_{γ}(C, ψ_{0}ϕϵ̄), I_{γ}(C, ϕ_{0}ψη̄)} ≥ w (†) Hence, γ is w-fold for the g_{2n}^{t} and G is contained in the fixed branch

contribution of the g_{2n}^{t}. In order to obtain the exactness statement, (∗), first observe that, if γ is a fixed branch of the g_{2n}^{t}, then, in particular, it belongs to (C ⊓ ϕ_{0}ψ_{0} = 0). Hence, it belongs either to (C ⊓ ϕ_{0} = 0) or (C ⊓ ψ_{0} = 0). Hence, it belongs to G. Now, using the fact that the original {g_{n}^{r}, g_{n}^{s}} had no fixed branch contribution, we can easily find ϕϵ̄_{0} and ψη̄_{0} with G disjoint from both (C ⊓ ϕϵ̄_{0} = 0) and (C ⊓ ψη̄_{0} = 0). Then, by the same argument (†), we obtain, for a branch γ of G;

I_{γ}(C, ψ_{0}ϕϵ̄_{0} + ϕ_{0}ψη̄_{0}) = w

hence, γ is counted w-times in C ⊓(ψ_{0}ϕϵ̄_{0} +ϕ_{0}ψη̄_{0} = 0) and, therefore, (∗) holds, as required. Now, as G had total multiplicity n, removing this fixed branch contribution from the g_{2n}^{t}, we obtain a g_{n}^{t}. We then claim that g_{n}^{r} ⊆ g_{n}^{t} and g_{n}^{s} ⊆ g_{n}^{t}, (∗∗). By Definition 4.14, it is sufficient to check that, if {W_{1}, W_{2}} are weighted sets appearing in {g_{n}^{r}, g_{n}^{s}}, defined by (C ⊓ϕϵ̄ = 0) and (C ⊓ψη̄ = 0), then they appear in the g_{n}^{t}. We clearly have that both ψ_{0}ϕϵ̄ and ϕ_{0}ψη̄ belong to Σ and the calculation (†) shows that;

C ⊓ (ψ_{0}ϕϵ̄ = 0) = W_{1} + G

C ⊓ (ϕ_{0}ψη̄ = 0) = W_{2} + G

Hence, the result (∗∗) follows after removing the fixing branch contribution G. The fact that r ≤ t and s ≤ t then follows easily from the definition of the dimension of a g_{n}^{r} and Theorem 3.3.

Now consider the case when the {g_{n}^{r}, g_{n}^{s}} are defined exactly by linear systems and have a fixed branch contribution. Let G_{1} ⊆ G and G_{2} ⊆ G be these fixed branch contributions and let G_{3} = G_{1} ∩ G_{2}. We claim that the fixed branch contribution of the g_{2n}^{t} defined by Σ, as given above, in this case is exactly G_{3} + G. The proof is similar to the above and left to the reader. Now, removing the fixed branch contribution G, we obtain a series g_{n}^{t} with fixed branch contribution G_{3}. A similar proof to the above, left to the reader, shows that this g_{n}^{t} contains the original series {g_{n}^{r}, g_{n}^{s}}. Finally, we need to consider the case when the {g_{n}^{r}, g_{n}^{s}} are defined, after removing some fixed branch contribution from linear series. Let G_{1} and G_{2}, with total multiplicity r_{1} and r_{2}, be these fixed branch contributions and let {g_{n+r_{1}}^{r}, g_{n+r_{2}}^{s}} be the series obtained from adding these fixed branch contributions to {g_{n}^{r}, g_{n}^{s}}. In this case, the linear system Σ, as given above, defines a g_{2n+r_{1}+r_{2}}^{t}. We claim that the weighted set G∪G_{1}∪G_{2}, of total multiplicity (n+r_{1}+r_{2}), is contained in the fixed branch contribution of this series. This follows from a similar calculation, using the method above, the details are left to the reader. Removing this weighted set from the g_{2n+r_{1}+r_{2}}^{t}, we obtain a g_{n}^{t} and a similar calculation shows that this contains the original {g_{n}^{r}, g_{n}^{s}}, again the details are left to the reader. □

As a corollary, we have;

Theorem 4.16. Let a g_{n}^{r} be given on C, then there exists a unique g_{n}^{t} on C, with r ≤ t ≤ n, such that;

g_{n}^{r} ⊆ g_{n}^{t}

and, for any g_{n}^{s} such that g_{n}^{r} ⊆ g_{n}^{s}, we have that;

g_{n}^{s} ⊆ g_{n}^{t}

Proof. By Lemma 3.16, we can find r ≤ t ≤ n and a g_{n}^{t} on C, with g_{n}^{r} ⊆ g_{n}^{t} and t maximal with this property. If g_{n}^{r} ⊆ g_{n}^{s}, then {g_{n}^{s}, g_{n}^{t}} would contain a common weighted set. By Theorem 4.15, we could then find t′ ≤ n such that s ≤ t′, t ≤ t′ and g_{n}^{s} ⊆ g_{n}^{t′}, g_{n}^{t} ⊆ g_{n}^{t′}. If g_{n}^{s} ⊊ g_{n}^{t}, then, by elementary dimension considerations, we would have that t < t′ ≤ n and g_{n}^{r} ⊂ g_{n}^{t′}, contradicting maximality of t. Hence, g_{n}^{s} ⊆ g_{n}^{t}. The uniqueness statement also follows from a similar amalgamation argument, using Theorem 4.15.

□

We can then make the following definition;

Definition 4.17. We call a g_{n}^{r} on C complete if it cannot be strictly contained in a g_{n}^{t} of greater dimension. If G is any weighted set on C of total multiplicity n, then we define |G| to be the unique complete g_{n}^{t} to which G belongs.

We then have that;

Theorem 4.18. Let G be a weighted set on C, then, G ≡ G′ if and only if G′ belongs to |G|. In particular, G ≡ G′ if and only if |G| = |G′|.

Proof. The proof of the first part of the theorem is quite straightforward. By definition, if G′ belongs to |G|, then G ≡ G′. Conversely, if G′ ≡ G, then, by Definition 4.6, we can find a g_{n}^{1}, containing the given weighted sets G and G′. By Theorem 4.16, we can find a unique complete g_{n}^{t} on C, with 1 ≤ t ≤ n, such that g_{n}^{1} ⊆ g_{n}^{t}. As G belongs to this g_{n}^{t} as a weighted set, it follows by Definition 4.17 that |G| = g_{n}^{t}. Hence, G′ belongs to |G| as required. For the second part, if G ≡ G′, then, by the first part, G′ belongs to |G|. It follows immediately from Definition 4.17 and Theorem 4.16, that |G| ⊆ |G′|. Reversing this argument, we have that |G′| ⊆ |G|, hence |G| = |G′| as required. Conversely, if |G| = |G′|, then clearly G ≡ G′ by Definition 4.6. □

We now make the following definition;

Definition 4.19. Linear System of a Weighted Set

Let G be a weighted set on a projective algebraic curve C, then we define the Riemann-Roch space L(C, G) or L(G) to be the vector space defined as;

{g ∈ L(C)^{∗} : div(g) + G ≥ 0} ∪ {0}

where div(g) was defined in Definition 4.9.

Remarks 4.20. That L(G) defines a vector space follows easily from Lemma 4.10, the fact that, for non-constant rational functions {f, g, f+ g} ⊂ L(C) and a branch γ of C, we have that;

ord_{γ}(f + g) ≥ min{ord_{γ}(f), ord_{γ}(g)}, (∗)

where, for this remark only, ord_{γ} is counted negatively if val_{γ} is infinite, and an argument on constants, (∗∗). We now give a brief proof of (∗);

We just consider the following 2 cases;

Case 1. val_{γ}(f) < ∞ and val_{γ}(g) < ∞

We then have, substituting the relative parametrisations, that;

f ∼ c + c_{1}t^{m} + … and g ∼ d + d_{1}t^{n} + …, where ord_{γ}(f) = m ≥ 1, ord_{γ}(g) = n ≥ 1 and {c_{1}, d_{1}} ⊂ L are non-zero. Then;

f + g ∼ (c + d) + c_{1}t^{m} + d_{1}t^{n} + …

If (f +g)−(c+d) ≡ 0, as an algebraic power series in L[[t]], then (f + g) = (c + d) as a rational function on C, contradicting the assumption. Hence, we obtain that ord_{γ}(f + g) = min{ord_{γ}(f), ord_{γ}(g)}, if m ≠ n or m = n and c_{1} + d_{1} ≠ 0, and ord_{γ}(f + g) > min{ord_{γ}(f), ord_{γ}(g)} otherwise. Hence, (∗) is shown in this case.

Case 2. val_{γ}(f) = val_{γ}(g) = ∞

We then have that;

f ∼ c_{1}t^{−m} + … and g ∼ d_{1}t^{−n} + …, where ord_{γ}(f) = −m ≤ −1, ord_{γ}(g) = −n ≤ −1 and {c_{1}, d_{1}} ⊂ L are non-zero. Then;

f + g ∼ c_{1}t^{−m} + d_{1}t^{−n} + …

By the assumption that f + g is not a constant, if m = n and c_{1} + d_{1} = 0, we must have higher order terms in t in the Cauchy series for (f + g), hence ord_{γ}(f + g) > min{ord_{γ}(f), ord_{γ}(g)}. Otherwise, we have that ord_{γ}(f + g) = min{ord_{γ}(f), ord_{γ}(g)}, hence (∗) is shown in this case as well.

The remaining cases are left to the reader. One should also consider the case of constants, (∗∗). Technically, one cannot define ord_{γ} for a constant in L. However, we did, by convention, define div(c) = 0, for c ∈ L^{∗}, in Remarks 4.5.

We now show the following;

Lemma 4.21. For a weighted set G, dim(L(G)) = t + 1, where t is given in Definition 4.17. In particular, L(G) is finite dimensional.

Proof. Let t be given by Definition 4.17. If t = 0, then G = (0) and L(G) = L. This follows easily from the well known fact that the only regular functions on a projective algebraic curve are the constants (see, for example, [10], p59). In this case, we then have that dim(L(G)) = 1, as required. Otherwise, let t ≥ 1 be given as in Definition 4.17, with the unique complete g_{n}^{t} containing G. After adding some fixed branch contribution W, we can find a linear system Σ, having finite intersection with C, with basis {ϕ_{0}, …, ϕ_{j}, …, ϕ_{t}} defining this g_{n}^{t}. Moreover, we may assume that C ⊓ ϕ_{0} = G ∪ W, (∗). Let {f_{1}, …, f_{j}, …, f_{t}} be the sequence of rational functions on C defined by f_{j} = \frac{ϕ_{j}}{ϕ_{0}} . We claim that; div(f_{j}) + G ≥ 0, for 1 ≤ j ≤ t (∗∗)

In order to show (∗∗), it is sufficient to prove that, for a branch γ with val_{γ}(f_{j}) = ∞, we have that γ belong to G and, moreover, that γ is counted at least ord_{γ}(f_{j}) times in G. Let Σ_{j} be the pencil of forms defined by (ϕ_{j} − λϕ_{0})_{λ∈P^{1}}. By the proof of Lemma 4.4, we have that (f_{j} = ∞) is defined by (C ⊓ ϕ_{0}), after removing the fixed branch contribution of this pencil. By (∗) and the fact that the fixed branch contribution of Σ_{j} includes W, we have that (f_{j} = ∞) ⊆ G. Hence, (∗∗) is shown as required. By Definition 4.19, we then have that f_{j} belongs to L(G). We now claim that there do not exist constants {c_{0}, …, c_{j}, …, c_{t}} ⊂ L such that;

c_{0} + c_{1}f_{1} + … + c_{j}f_{j} + … + c_{t}f_{t} = 0 (∗ ∗ ∗)

as rational functions on C. If so, we would have that;

c_{0}ϕ_{0} + c_{1}ϕ_{1} + … + c_{j}ϕ_{j} + … + c_{t}ϕ_{t}

vanished identically on C, contradicting the fact that Σ has finite intersection with C. Hence, by (∗ ∗ ∗), {1, f_{1}, …, f_{t}} ⊂ L(G) are linearly independent and dim(L(G)) ≥ t + 1. Conversely, suppose that dim(L(G)) ≥ k + 1, then we can find {1, f_{1}, …, f_{j}, …, f_{k}} ⊂ L(G) which are linearly independent, (†). By the usual method of equating denominators, we can find algebraic forms {ϕ_{0}, …, ϕ_{k}} of the same degree, such that f_{j} is represented by \frac{ϕ_{j}}{ϕ_{0}} , for 1 ≤ j ≤ k. Let Σ be the linear system defined by this sequence of forms. By (†), Σ has finite intersection with C. Let W, having total multiplicity n′, be the fixed branch contribution of this system and let (C ⊓ ϕ_{0}) = G_{0} ∪ W. We claim that G_{0} ⊆ G, (††). Suppose not, then there exists a branch γ with I_{γ}^{Σ,mobile}(C, ϕ_{0}) = s, where γ is counted strictly less than s-times in G. By the definition of I_{γ}^{Σ,mobile}, we can find a form ϕ_{λ} belonging to Σ, distinct from ϕ_{0}, witnessing this multiplicity. Consider the pencil Σ_{λ} defined by (ϕ_{λ}−μϕ_{0})_{μ∈P^{1}}. We then clearly have that I_{γ}^{Σ_{λ},mobile}(C, ϕ_{0}) = s as well, (†††). Let f_{λ} = \frac{ϕ_{λ}}{ϕ_{0}} . By the proof of Lemma 4.4, we have that (f_{λ} = ∞) is defined by (C ⊓ ϕ_{0}), after removing the fixed branch contribution of Σ_{λ}. By (†††), it follows that the branch γ is counted stimes in (f_{λ} = ∞) and therefore div(f_{λ})+G ≱ 0. However, f_{λ} is a linear combination of {1, …, f_{k}}, hence f_{λ} ∈ L(G), which is a contradiction. Hence, (††) is shown. Now, consider the g_{n}^{k} defined by Σ. Let W′ be the weighted set G\G_{0} of total multiplicity n′′. By Lemma 3.13, we can add the weighted set W′ to the g_{n}^{k} and obtain a g_{n+n′′}^{k} with fixed branch contribution W′ ∪ W. Now, removing the fixed branch contribution W from this g_{n+n′}^{k}, we obtain a g_{n+n′′−n′}^{k} containing G exactly as a weighted set. It follows, from Definition 4.17, that k ≤ t. Hence, in particular, dim(L(G)) is finite and dim(L(G) ≤ t + 1. Therefore, the lemma is proved.

□

We now extend the notion of linear equivalence to include virtual, or non-effecive, weighted sets.

Definition 4.22. We define a generalised weighted set G on C to be a linear combination of branches;

n_{1}γ_{p_{1}}^{j_{1}} + … + n_{r}γ_{p_{r}}^{j_{r}}

where {n_{1}, …, n_{r}} belong to Z. If {n_{1}, …, n_{r}} belong to Z_{≥0}, we call the weighted set effective. Otherwise, we call the weighted set virtual. We define n = n_{1} + … + n_{r} to be the total multiplicity or degree of G.

Remarks 4.23. It is an easy exercise to see that there exist well defined operations of addition and subtraction on generalised weighted sets. It is also easy to check that any generalised weighted set G may be written uniquely as G_{1}−G_{2}, where {G_{1}, G_{2}} are disjoint effective weighted sets.

Definition 4.24. Let A and B be generalised weighted sets on C of the same total multiplicity. Let {A_{1}, A_{2}} and {B_{1}, B_{2}} be the unique effective weighted sets, as given by the previous remark. Then we define;

(A_{1} − A_{2}) ≡ (B_{1} − B_{2}) iff (A_{1} + B_{2}) ≡ (B_{1} + A_{2})

and;

A ≡ B iff (A_{1} − A_{2}) ≡ (B_{1} − B_{2})

Remarks 4.25. Note that if {A′_{1}, A′_{2}} and {B′_{1}, B′_{2}} are any effective weighted sets such that;

A = A′_{1} − A′_{2} and B = B′_{1} − B′_{2}

then A ≡ B iff A′_{1} + B′_{2} ≡ B′_{1} + A′_{2}

48 TRISTRAM DE PIRO

The proof is just manipulation of effective weighted sets. We clearly have that;

A_{1} + A′_{2} = A′_{1} + A_{2} and B_{1} + B′_{2} = B′_{1} + B_{2} (∗) We then have; A ≡ B iff A_{1} + B_{2} ≡ B_{1} + A_{2}(Definition 4.24)

iff A_{1} + A′_{2} + B_{2} ≡ B_{1} + A_{2} + A′_{2} (Theorem 4.13)

iff A′_{1} + A_{2} + B_{2} ≡ B_{1} + A_{2} + A′_{2} (by (*))
iff A′_{1} + B_{2} ≡ B_{1} + A′_{2} (Theorem 4.13)
iff A′_{1} + B_{2} + B′_{1} ≡ B_{1} + B′_{1} + A′_{2} (Theorem 4.13)
iff A′_{1} + B_{1} + B′_{2} ≡ B_{1} + B′_{1} + A′_{2} (by (*))
iff A′_{1} + B′_{2} ≡ B′_{1} + A′_{2} (Theorem 4.13)

We then have;

Theorem 4.26. Transitivity of Linear Equivalence

Let C′ be an algebraic curve. If A, B, C are generalised weighted sets on C′ of the same total multiplicity, then, if A ≡ B and B ≡ C, we must have that A ≡ C. Proof. Let {A_{1}, A_{2}}, {B_{1}, B_{2}} and {C_{1}, C_{2}} be the effective weighted sets as given by Remarks 4.23. Then, by Definition 4.24, we have that;

(A_{1} + B_{2}) ≡ (B_{1} + A_{2}) and (B_{1} + C_{2}) ≡ (C_{1} + B_{2}) By Theorem 4.13, we have that; (A_{1} + B_{1} + B_{2} + C_{2}) ≡ (C_{1} + B_{1} + B_{2} + A_{2}) It then follows, by Definition 4.6, that there exists a g_{n}^{1}, containing

(A_{1} + B_{1} + B_{2} + C_{2}) and (C_{1} + B_{1} + B_{2} + A_{2}) as weighted sets. Clearly (B_{1} + B_{2}) is contained in the fixed branch contribution of this g_{n}^{1}. Removing this fixed branch contribution, we obtain;

A_{1} + C_{2} ≡ C_{1} + A_{2} By Definition 4.24, we then have that A ≡ C as required.

□

It follows immediately from Theorem 4.12 and Theorem 4.26 that;

Theorem 4.27. Let C be a projective algebraic curve, then ≡ is an equivalence relation on generalised weighted sets for C of a given total multiplicity.

Remarks 4.28. Again, the definition of linear equivalence that we have given for generalised weighted sets on a smooth projective algebraic curve C is equivalent to the modern definition for divisors. More precisely, we have;

Modern Definition; Let A and B be divisors on a smooth projective algebraic curve C, then A ≡ B iff A−B = div(g), for some g ∈ L(C)^{∗}.

See, for example, p161 of [10] for relevant definitions and notation. In order to show that our definition is the same, use Remarks 4.8 and the following simple argument;

A ≡ B as generalised weighted sets iff A_{1} + B_{2} ≡ B_{1} + A_{2}

where {A_{1}, A_{2}, B_{1}, B_{2}} are the effective weighted sets given by Definition 4.24. Then;

A_{1} + B_{2} ≡ B_{1} + A_{2} iff (A_{1} + B_{2}) − (B_{1} + A_{2}) = div(g) (g ∈ L(C)^{∗})

by Remarks 4.8, where div(g) is the modern definition. By a straightforward calculation, we have that;

(A_{1} + B_{2}) − (B_{1} + A_{2}) = A − B as divisors or generalised weighted

sets.

Hence, the notions of equivalence coincide.

We also have;

Theorem 4.29. Linear Equivalence Preserved by Addition

Let C′ be a projective algebraic curve and suppose that {A, B, C, D} are generalised weighted sets on C′ with;

A ≡ B and C ≡ D then;

A + C ≡ B + D

Proof. Let {A_{1}, A_{2}}, {B_{1}, B_{2}}, {C_{1}, C_{2}} and {D_{1}, D_{2}} be effective weighted sets as given by Remarks 4.23 Then, by Definition 4.24, we have that;

A_{1} + B_{2} ≡ B_{1} + A_{2} and C_{1} + D_{2} ≡ D_{1} + C_{2}

Hence, by Theorem 4.13;

A_{1} + B_{2} + C_{1} + D_{2} ≡ B_{1} + A_{2} + D_{1} + C_{2} (∗)

We clearly have that;

A + C = (A_{1} + C_{1}) − (A_{2} + C_{2}) and B + D = (B_{1} + D_{1}) − (B_{2} + D_{2})

as an identity of generalised weighted sets. Moreover, as (A_{1} + C_{1}), (A_{2} + C_{2}), (B_{1} + D_{1}) and (B_{2} + D_{2}) are all effective, we can apply Remarks 4.25 and (∗) to obtain the result. □

We now make the following definition;

Definition 4.30. Let G be a generalised weighted set on a projective algebraic curve C, then we define |G| to be the collection of generalised weighted sets G′ with G′ ≡ G. We define order(|G|) to be the total multiplicity (possibly negative) of any generalised weighted set in |G|.

Remarks 4.31. If G is an effective weighted set, the collection defined by Definition 4.30 is not the same as the collection given by Definition 4.17, as it includes virtual weighted sets. Unless otherwise stated, we will use Definition 4.17 for effective weighted sets. This convention is in accordance with the Italian terminology.

We now show that the notions of linear equivalence introduced in this section are birationally invariant;

Theorem 4.32. Let Φ : C_{1} ↭ C_{2} be a birational map. Let A and B be generalised weighted sets on C_{2}, with corresponding generalised weighted sets [Φ]^{∗}A and [Φ]^{∗}B on C_{1}. Then A ≡ B, in the sense of either Definition 4.6 or 4.24, iff [Φ]^{∗}A ≡ [Φ]^{∗}B.

Proof. Suppose that A ≡ B in the sense of Definition 4.6. Then, there exists a g_{n}^{r} on C_{2} containing A and B as weighted sets. By Theorem 3.14, there exists a corresponding g_{n}^{r} on C_{1}, containing [Φ]^{∗}A and [Φ]^{∗}B as weighted sets. Hence, again by Definition 4.6, [Φ]^{∗}A ≡ [Φ]^{∗}B. The converse is similar, using [Φ^{−1}]^{∗}. If A ≡ B in the sense of Definition 4.24, then the same argument works.

□

As a result of this theorem, we introduce the following definition;

Definition 4.33. Let Φ : C_{1} ↭ C_{2} be a birational map. Then, given a generalised weighted set A on C_{2}, we define;

[Φ]^{∗}|A| = |[Φ]^{∗}A|

where, in the case that A is effective, |A| can be taken either in the sense of Definition 4.17 or Definition 4.30.

Remarks 4.34. The definition depends only on the complete series |A|, rather than its particular representative A. This follows immediately from Definition 4.17, Definition 4.30 and Theorem 4.32.

We finally introduce the following definition;

Definition 4.35. Summation of Complete Series

Let A and B be generalised weighted sets, defining complete series |A| and |B|, in the sense of Definition 4.30. Then, we define the sum;

|A| + |B|

to be the complete series, in the sense of Definition 4.30, containing all generalised weighted sets of the form A′ + B′ with A′ ∈ |A| and B′ ∈ |B|. If A and B are effective weighted sets with |A|, |B| taken in the sense of Definition 4.17, then we make the same definition for the sum in the sense of Definition 4.17.

Remarks 4.36. This is a good definition by Theorem 4.13 and Theorem 4.29.

Definition 4.37. Difference of Complete Series

52 TRISTRAM DE PIRO

Let A and B be generalised weighted sets, defining complete series |A| and |B|, in the sense of Definition 4.30. Then, we define the difference;

|A| − |B| to be the complete series, in the sense of Definition 4.30, containing

all generalised weighted sets of the form A′ − B′ with A′ ∈ |A| and B′ ∈ |B|. If A and B are effective weighted sets with |A|, |B| taken in the sense of Definition 4.17, then we can in certain cases define a difference in the sense of Definition 4.17. (This is called the residual series, the reader can look at [9] for more details) Remarks 4.38. This is again a good definition, for generalised weighted sets {A, B}, it follows trivially from the previous definition and the fact that {A, −B} are also generalised weighted sets.

References

[1] S.S. Abhyankar, Algebraic Geometry for Scientists and Engineers, AMS Mathematical Surveys 35, (1990)

[2] R. Hartshorne, Algebraic Geometry, Springer (1977)

[3] D. Mumford. Red Book of Varieties and Schemes, Springer (1999)

[4] K. Peterzil and B. Zilber, Lecture Notes on Zariski Structures (1996)

[5] T. de Piro, A Non-Standard Bezout Theorem for Curves, available on my website http://www.magneticstrix.net, (2010), old version ”A Non-Standard Bezout Theorem” on AG/LO ArXiv (0406176), (2004).

[6] T. de Piro, A Theory of Branches for Algebraic Curves, available on my website http://www.magneticstrix.net, (2010), old version MODNET preprint server, (2006).

[7] T. de Piro, Infinitesimals in a Recursively Enumerable Prime Model, available on my website http://www.magneticstrix.net, (2010), also on LO ArXiv (0510412), (2005).

[8] T. de Piro, Zariski Structures and Algebraic Curves, available on my website http://www.magneticstrix.net, (2010), old version ”Zariski Structures and Algebraic Geometry” on AG ArXiv (0402301), (2004).

[9] F. Severi, Trattato di geometria algebrica, 1: Geometria delle serie lineari, Zanichelli, Bologna, (1926).

A THEORY OF DIVISORS FOR ALGEBRAIC CURVES 53

[10] I. Shafarevich, Basic Algebraic Geometry 1, Springer (1977).

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