A NON-STANDARD BEZOUT THEOREM FOR CURVES

TRISTRAM DE PIRO

Abstract. This paper provides a non-standard analogue of Bezout’s theorem for algebraic curves. We achieve this by showing that, in all characteristics, the notion of Zariski multiplicity coincides with intersection multiplicity when we consider the full families of projective degree d and degree e curves in P^{2}(L). The result is particularly interesting in that it holds even when we consider intersections at singular points of curves or when the curves contain non-reduced components.

The techniques of non-standard analysis, originally developed for the real numbers, were recently introduced by Zilber in the context of Zariski structures. This paper sets out to show that non-standard analysis can be useful in algebraic geometry, by providing a more geometric framework for understanding intersections of algebraic curves in the plane. We assume some familiarity with certain notions from algebraic and analytic geometry, as well as the material from Sections 1-5 of [6]. We summarise the relevant facts, for the proofs of the paper, in the following three sections;

1. Etale Morphisms and Algebraic Multiplicity

Definition 1.1. A morphism f of finite type between varieties X and Y is said to be etale if for all x ∈ X there are open affine neighborhoods U of x and V of f(x) with f(U) ⊂ V such that restricted to these neighborhoods the pull back on functions is given by the inclusion;

f^{∗} : L[V] → L[V]\frac{[x_{1},...,x_{n}]}{<f_{1},...,f_{n}>}

and det(\frac{∂f_{i}}{∂x_{j}})(x) ≠ 0 , (∗)

The coordinate free definition of etale is that f should be flat and unramified, where a morphism f is unramified if the sheaf of relative differentials Ω_{X/Y} = 0, clearly this last confition is satisfied using the condition (∗). If we tensor the exact sequence,

The author was supported by the William Gordon Seggie Brown research fellowship.

f^{∗}Ω_{Y} → Ω_{X} → Ω_{X/Y} → 0

with L(x) the residue field of x, we obtain an isomorphism

f^{∗}Ω_{Y} ⊗ L(x) → Ω_{X} ⊗ L(x).

Identifying Ω_{X} ⊗ L(x) with T_{x,X}^{∗} gives that

df : (m_{x}/m_{x}^{2})^{∗} → (m_{f(x)}/m_{f(x)}^{x})^{∗}

is an isomorphism of tangent spaces or dually f^{∗}(m_{f(x)}) = m_{x}. Call this property of etale morphisms (∗∗).

We will also require some facts about the etale topology on an algebraic variety Y, see [14] for more details. We consider a category Y_{et} whose objects are etale morphisms U → Y and whose arrows are Y -morphisms from U → V. This category has the following 2 desirable properties. First given y ∈ Y, the set of objects of the form (U, x) → (Y, y) form a directed system, namely (U, x) ⊂ (U′, x′) if there exists a morphism U → U′ taking x to x′. Secondly, we can take “intersections” of open sets U_{i} and U_{j} by considering U_{ij} = U_{i} ×_{Y} U_{j}; the projection maps are easily show to be etale and the composition of etale maps is etale, so U_{ij} → Y still lies in Y_{et}. If Y is an irreducible variety over L, then all etale morphisms into Y must come from reduced schemes of finite type over L, though they may well fail to be irreducible considered as algebraic varieties. Now we can define the local ring of Y in the etale toplogy to be;

O_{y,Y}^{∧} = lim_{→,y∈U}O_{U}(U)

As any open set U of Y clearly induces an etale morphism U →_{i} Y of inclusion, we have that O_{y,Y} ⊂ O_{y,Y}^{∧}. We want to prove that O_{y,Y}^{∧} is a Henselian ring and in fact the smallest Henselian ring containing O_{y,Y}. We need the following lemma about Henselian rings;

Lemma 1.2. Let R be a local ring with residue field k. Suppose that R satisfies the following condition;

If f_{1}, … f_{n} ∈ R[x_{1}, … x_{n}] and f̄_{1} … f̄_{n} have a common root ā in k^{n}, for which Jac(f̄)(ā) = (\frac{∂f̄_{i}}{∂x_{j}})_{ij}(ā) ≠ 0, then ā lifts to a common root in R^{n} (*).

Then R is Henselian.

It remains to show that O_{y,Y}^{∧} satisfies (∗).

Proof. Given f_{1,} … f_{n} satisfying the condition of (∗), we can assume the coefficients of the f_{i} belong to O_{U_{i}}(U_{i}) for covers U_{i} → Y; taking the intersection U_{1...i...n} we may even assume the coefficients define functions on a single etale cover U of Y. By the remarks above we can consider U as an algebraic variety over K, and even an affine algebraic variety after taking the corresponding inclusion. We then consider the variety V ⊂ U × A^{n} defined by Spec(\frac{R(U)[x_{1},...,x_{n}]}{f_{1},...f_{n}}). Letting u ∈ U denote the point in U lying over y ∈ Y, the residue of the coefficients of the f_{i} at u corresponds to the residue in the local ring R, which tells us exactly that the point (u, ā) lies in V. By the Jacobian condition, we have that the projection π : V → U is etale at the point (u, ā), and hence on some open neighborhood of (u, ā), using Nakayama’s Lemma applied to Ω_{V/U}. Therefore, replacing V by the open subset U′ ⊂ V gives an etale cover of U and therefore of Y, lying over y. Now clearly the coordinate functions x_{1}, … x_{n} restricted to U′ lie in O_{y,Y}^{∧} and lift the root ā to a root in O_{y,Y}^{∧}

□

We define the Henselization of a local ring R to be the smallest Henselian ring R′ ⊃ R, with R′ ⊂ F rac(R)^{alg}. We have in fact, see [14], that;

Theorem 1.3. Given an algebraic variety Y, O_{y,Y}^{∧} is the Henselization of O_{y,Y}

The following theorem requires some knowledge of Zariski structures, see sections 1-4 of [6], or section 2 of this paper.

Theorem 1.4. Zariski multiplcity is preserved by etale morphisms

Let π : X → Y be an etale morphism with Y smooth, then any (ab) ∈ graph(π) ⊂ X × Y 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 [13]; Fact 1.5. 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.

Using Lemma 4.6 of [6] 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).

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}.

We also review some facts about algebraic multiplicity and show that algebraic multiplicity is preserved by etale morphisms.

Definition 1.6. Given projective varieties X_{1}, X_{2} and a finite morphism f : X_{1} → X_{2}, the algebraic multiplicity mult_{af(a)}^{alg}(X_{1}/X_{2}) of f at a ∈ X_{1} is length(O_{a,X_{1}}/f^{∗}m_{f(a)}) where m_{f(a)} is the maximal ideal of the local ring O_{f(a)}.

Remarks 1.7. Note that this is finite, by the fact that finite morphisms have finite fibres and the ring O_{a,X_{1}}/f^{∗}m_{f(a)} is a localisation of the fibre f^{−1}(f(a)) ≅ R(f^{−1}(U)) ⊗_{R(U)} L ≅ R(f^{−1}(U))/m_{f(a)} where U is an affine subset of X_{2} containing f(a).

We now have the following;

Theorem 1.8. Algebraic multiplicity is preserved by etale morphisms;

Given finite morphisms f : X_{3} → X_{2} and g : X_{2} → X_{1} with f etale. If a ∈ X_{3}, then mult_{a,gf(a)}^{alg}(X_{3}/X_{1}) = mult_{f(a),gf(a)}^{alg}(X_{2}/X_{1}).

Proof. This result is essentially given in [15]. Let O_{f(a),X_{2}}^{∧} be the Henselisation of the local ring at f(a). By base change, we have an etale morphism f′ : X′ = X_{3} ×_{X_{2}} Spec(O_{f(a)}^{∧}) → Spec(O_{f(a)}^{∧}). By the definition of an etale morphism given above, we may write this cover locally in the form Spec(O_{f(a)}^{∧} \frac{[x_{1},...,x_{n}]}{f_{1},...,f_{n}}), with det(\frac{∂f_{i}}{∂x_{j}}) ≠ 0 at each closed point in the fibre over f(a). At the closed point a, let a_{i} be the residues of the x_{i} in L, then we have that (a_{1}, … a_{n}) is a common root for {f̄_{1}, …, f̄_{n}} where f̄_{i} is obtained by reducing f_{i} with respect to the maximal ideal m_{f(a),X_{2}} of O_{f(a)}^{∧}. As O_{f(a)}^{∧} is Henselian, by the above, and the determinant condition, we can lift the roots a_{i} to roots α_{i} of the f_{i} in O_{f(a)}^{∧}. We therefore obtain a subscheme Z = Spec(O_{f(a)}^{∧} \frac{[x_{1},...,x_{n}]}{<x_{1}−α_{1},...,x_{n}−α_{n}} >) of X′ which is isomorphic to Spec(O_{f(a)}^{∧}) under the restriction of f. Let Q be the O_{X′} ideal defining Z, we then have that m_{a,X′} = f^{∗}m_{f(a),X_{2}} ⊕ Q_{a}. As f is etale, by (∗∗) after definition 1.1 above, m_{a,X′} = f^{∗}m_{f(a),X_{2}}, therefore Q_{a} = 0 and by Nakayama’s lemma Q = 0 in an open neighborhood of a in X′. This gives that Z = X′ in an open neighborhood of a. Hence we obtain the sequence O_{f(a),X_{2}} →_{f^{∗}} O_{a,X_{3}} →_{i^{∗}} O_{a,X′} (***) where the map i^{∗}f^{∗} is the inclusion of O_{f(a),X_{2}} inside O_{f(a),X_{2}}^{∧}. Now if n ⊂ m_{f(a),X_{2}} is the pullback g^{∗}m_{gf(a),X_{1}}, we have that length(O_{f(a),X_{2}}/n) = length(O_{f(a),X_{2}}^{∧}/n), hence the result follows by (∗ ∗ ∗) as required.

□

2. Zariski Multiplicity

We work in the context of Theorem 3.3 in [6]. Namely, W (we used the notation V in [6]) will denote a smooth projective variety defined over an algebraically closed field L, considered as a Zariski structure with closed sets given by algebraic subvarieties defined over L. All notions connected to the definition of Zariski multiplicity will come from a fixed specialisation map π : W (K_{ω}) → W (L) where K_{ω} denotes a ”universal” algebraically closed field containing L = K_{0}. We consider D a smooth subvariety of some cartesian power W^{m} and a finite cover, with respect to projection onto the first coordinate, F ⊂ D × W^{k}, all defined over L (*). This allows us to make sense of Zariski multiplicity. In general, we can move freely between Zariski structure notation and algebraic geometry notation. Clearly (∗) makes sense algebraically. Conversely, if X and Y denote fixed projective varieties defined over L with Y smooth and a finite morphism f : X → Y over L is given , then we can reduce to the situation of (∗) by taking F to be graph(f) ⊂ X × Y with the projection map onto the second factor and W to be the corresponding projective space P^{n}(L) where X, Y ⊂ P^{n}(L). We can even take W to be the 1-dimensional Zariski structure P^{1}(L) by using the embedding of P^{n}(L) into the N’th Cartesian power of P^{1}(L) for sufficiently large N.

We use the definition of Zariski multiplicity for irreducible finite covers given in 4.1 of [6]. We will also require the following generalisation;

Definition 2.1. Let F ⊂ D×W^{k} be an equidimensional, finite cover of smooth D, with irreducible components C_{1}, …, C_{n}. Then for (ab) ∈ F, we define Mult_{ab}(F/D) = ∑_{(ab)∈C_{i}} Mult_{ab}(C_{i}/D).

Clearly this is well defined using the definition of Zariski multiplicity for irreducible covers. However, until Lemma 2.9, the assumption that F is irreducible will be in force.

Lemma 2.2. Zariski multiplicity is multiplicative over composition

Suppose that F_{1}, F_{2} and F_{3} are smooth, irreducible, with F_{2} ⊂ F_{1} × W^{k} and F_{3} ⊂ F_{2} × W^{l} finite covers. Let (abc) ∈ F_{3} ⊂ F_{1} × W^{k} × W^{l}. Then mult_{abc}(F_{3}/F_{1}) = mult_{ab}(F_{2}/F_{1})mult_{abc}(F_{3}/F_{2}).

Proof. To see this, let m = mult_{ab}(F_{2}/F_{1}) and n = mult_{abc}(F_{3}/F_{2}). Choose a′ ∈ V_{a} ∩ F_{1}(K_{ω}) generic over L. By definition, we can find distinct b_{1} … b_{m} in W^{k}(K_{ω}) ∩ V_{b} such that F_{2}(a′, b_{i}) holds. As F_{2} is a finite cover of F_{1}, we have that dim(a′b_{i}/L) = dim(a′/L) = dim(F_{1}) = dim(F_{2}), so each (a′b_{i}) ∈ V_{ab}∩F_{2} is generic over L. Again by definition, we can find distinct c_{i1} … c_{in} in W^{l}(K_{ω})∩V_{c} such that F_{3}(a′b_{i}c_{ij}) holds. Then the mn distinct elements (a′b_{i}c_{ij}) are in V_{abc}, so by definition of multiplicity mult_{abc}(F_{3}/F_{1}) = mn as required.

□

Lemma 2.3. Let hypotheses be as in the above lemma with the extra condition that the cover F_{3}/F_{2} is etale. Then for (abc) ∈ F_{3}, mult_{abc}(F_{3}/F_{1}) = mult_{ab}(F_{2}/F_{1})

Proof. This is an immediate consequence of Lemma 2.2 and Theorem 1.4.

□

Lemma 2.4. Zariski multiplicity is summable over specialisation

Suppose that F ⊂ D×W^{k} is a finite irreducible cover with D smooth. Suppose (ab) ∈ F, a′ ∈ V_{a} ∩ D and a′′ ∈ V_{a′} ∩ D with a′′ generic over L. Then

Mult_{ab}(F/D) = Σ_{b′∈V_{b}∩F (a′y)}Mult_{a′b′}(F/D)

Proof. Suppose F (a′′b_{1}), … F (a′′b_{n}) hold with b_{i} ∈ V_{b}, so {b_{1}, …, b_{n}} witness the fact that Mult_{ab}(F/D) = n. Write {b_{1}, … b_{n}} as {b_{11}, …, b_{1m_{1}}, b_{21}, …, b_{2m_{2}}, …, b_{i1}, … b_{ij}, …, b_{im_{i}}, …, b_{nm_{n}}} (*), where b_{ij} maps to a_{i} in the specialisation taking a′′ to a′. To prove the lemma, it is sufficient to show that F (a′y) ∩ V_{b} = {a_{1}, …, a_{n}} and Mult_{(a′a_{i})}(F/D) = m_{i}. The second statement just follows from the fact that a′′ is generic in D over L in V_{a′}. To prove the first statement, suppose we can find a_{n+1} with F (a′a_{n+1}) and a_{n+1} ∈ V_{b} but a_{n+1} ∉ {a_{1}, … a_{n}}. By Theorem 3.3 in [6], we can find c with F (a′′c) and (a′′c) specialising to (a′a_{n+1}). As a_{n+1} ∈ V_{b}, (a′a_{n+1}) specialises to (ab), hence so does (a′′c). Therefore, c must witness the fact that Mult_{ab}(F/D) = n and appear in the set {b_{1}, …, b_{n}}. This clearly contradicts the arrangement of {b_{1}, …, b_{n}} given in (∗).

□

Definition 2.5. Let F ⊂ U × V × W^{k} be an irreducible finite cover of U × V with U and V smooth.

Given (u, v, x) ∈ F we define;

LeftMult_{u,v,x}(F/D) = Card(V_{x} ∩ F (u′, v)) for u′ ∈ V_{u} ∩ U generic over L.

RightMult_{u,v,x}(F/D) = Card(V_{x} ∩ F (u, v′)) for v′ ∈ V_{v} ∩ V generic over L.

We first show that both left and right multiplicity are well defined. In order to see this, observe that the fibres F (u, V) and F (U, v) are finite covers of V and U respectively with U and V smooth. Moreover, the fibres F (u, V) and F (U, v) are equidimensional covers of V and U respectively. In order to see this, as U is smooth, it satisfies the presmoothness axiom with the smooth projective variety W^{k} given in Definition 1.1 of [6]. The fibre F (u, V) = F ∩ (W^{k} × {u} × V). By presmoothness, each irreducible component of the intersection has dimension at least dim(F) + dim(W^{k} × V) − dim(U × V × W^{k}) = dim(F) − dim(U) = dim(V). As F (u, V) is a finite cover of V, it has exactly this dimension. Now we can use the definition of Zariski multiplicity given in Definition 2.1.

We then claim the following;

Lemma 2.6. Factoring Multiplicity

In the situation of the above definition, we have that;

Mult_{u,v,x}(F/U ×V) = Σ_{x′∈(V_{x}∩F (y,u′,v))}RightMult_{x′,u′,v}(F/U ×V) for u′ generic in U over L.

Mult_{u,v,x}(F/U × V) = Σ_{x′∈(V_{x}∩F (y,u,v′))}LeftMult_{x′,u,v′}(F/U × V) for v′ generic in V over L.

Proof. We just prove the first statement, the proof of the second is apart from notation identical. By the construction in Section 2 and Lemma 3.2 of [6], we can choose algebraically closed fields L = K_{0} ⊂ K_{n_{1}} ⊂ K_{n_{2}} ⊂ K_{ω}, and tuples u′ ∈ K_{n_{1}}, v′ ∈ K_{n_{2}} such that u′ is generic in U over L, v′ is generic in V over K_{n_{1}} with specialisations π_{1} : P^{n}(K_{n_{1}}) → P^{n}(L) and π_{2} : P^{n}(K_{n_{2}}) → P^{n}(K_{1}) such that π_{2}(u′v′) = (u′v) and π_{1}(u′v) = (uv). Now dim(u′v′/L) = dim(v′/L(u′)) + dim(u′/L) = dim(V) + dim(U), hence u′v′ is generic in U × V over L. Therefore Mult_{u,v,x} = Card(V_{x}∩F (u′v′)). Let S = {y_{11}, …, y_{1m_{1}}, …, y_{ij_{i}}, …, y_{n1}, …, y_{nm_{n}}} be distinct elements in V_{x} ∩ W^{k} witnessing this multiplicity such that for 1 ≤ j_{i} ≤ m_{i}, π_{2}(y_{ij_{i}}) = z_{i} ∈ V_{x} ∩ W^{k}. It is sufficient to show that RightMult_{u′v,z_{i}}(F/U × V) = m_{i} and {z_{1}, … z_{n}} enumerates V_{x} ∩ F (y, u′, v). The first statement follows as v′ ∈ V_{v} ∩ V is generic in V over L(u′). For the second statement, suppose that we can find z_{n+1} ∈ V_{x} ∩ F (y, u′, v) with z_{n+1} ∉ {z_{1}, … z_{n}}. Consider F (u′, V) as a finite cover of V, defined over L(u′), so by the above F (u′, V) is an equidimensional finite cover of V. Then, as v′ was chosen to be generic in V over L(u′), choosing an irreducible component of F (u′, V) passing through (z_{n+1}, u′v), by the lifting result of Theorem 3.3 in [6], we can find y_{n+1} ∈ V_{z_{n+1}} ∩ W^{k} such that F (y_{n+1}, u′, v′). Clearly, y_{n+1} ∈ S which contradicts the definition of S.

□

Theorem 3.3 of [6] does not hold in the case when D fails to be smooth. However, in the case of etale covers, we still have the following result;

Lemma 2.7. Lifting Lemma for Etale Covers

Let F ⊂ D × W^{k} be an etale cover of D defined over L, with the projection map denoted by f. Then given a ∈ D, (ab) ∈ F and a′ ∈ V_{a} ∩ D generic over L, we can find b′ ∈ V_{b} such that F (a′, b′) holds. Moreover b′ is unique, hence Mult_{ab}(F/D) = 1. Moreover, in the situation of Lemma 2.3, without requiring that F_{2} is smooth, we have that for (abc) ∈ F_{3}, mult_{abc}(F_{3}/F_{1}) = mult_{ab}(F_{2}/F_{1}).

Proof. Using the definition of etale given in section 1 above, we can assume that the cover is given algebraically in the form f^{∗} : L[D] → L[D]\frac{[x_{1},...,x_{n}]}{f_{1},...,f_{n}} with det(\frac{∂f_{i}}{∂x_{j}})_{ij}(x) ≠ 0 for all x ∈ F. So we can present the cover in the form f_{1}(x, y) = 0, f_{2}(x, y) = 0, …, f_{n}(x, y) = 0, with y in D and x in A^{n}(L). Let L_{m} be the algebraic closure of the field generated by L and ḡ(a) where ḡ is a tuple of functions defining D locally. Consider the system of equations f_{1}(x, a) = f_{2}(x, a) = … = f_{n}(x, a) = 0 defined over L_{m}. Then this system is solved by b in L_{m} with the property that det(\frac{∂f_{i}}{∂x_{j}})_{ij}(b) ≠ 0 (*). Now suppose that a′ ∈ V_{a} ∩ D is chosen to be generic over L. By the construction given in Lemma 2.2 of [6], we may assume that a′ lies in L_{s}[[t^{1/r}]], the formal power series in the variable t^{1/r} for some algebraically closed field L_{s} extending L_{m}. This is a henselian ring, hence if we consider the system of equations f_{1}(x, a′) = f_{2}(x, a′) = … = f_{n}(x, a′) = 0 with coefficients in L_{s}[[t^{1/r}]], by the fact that the system specialises to a solution in L_{s} with the condition (*) we can find a solution b′ in L_{s}[[t^{1/r}]]. Then (a′b′) lies in F and by construction b′ ∈ V_{b}. The uniqueness result follows from the proof of Theorem 1.4. For the last part, suppose that mult_{ab}(F_{2}/F_{1}) = n, then we can find a′ ∈ V_{a} ∩ F_{1} generic over L and {b_{1}, … b_{n}} ∈ V_{b} ∩ W^{k} distinct such that F (a′, b_{i}) holds. Each (a′b_{i}) is generic in F_{2} over L, hence by the previous part of the lemma, we can find a unique c_{i} ∈ V_{c} ∩ W^{l} such that F_{3}(a′b_{i}c_{i}) holds. This show that mult_{abc}(F_{3}/F_{1}) = n as required.

□

Lemma 2.8. Lifting Lemma for Etale Covers with Right(Left) Multiplicity

Let hypotheses be as in Lemma 2.2, with the additional assumption that F_{1} = U × V, F_{2} is a smooth irreducible cover of F_{1} and F_{3} is an irreducible etale cover of F_{2}. Then, with notion as in Definition 2.5, given (uvbc) ∈ F_{3}, RightMult_{uvbc}(F_{3}/F_{1}) = RightMult_{uvb}(F_{2}/F_{1}). Similarly for left multiplicity.

Proof. Suppose that RightMult_{uvb}(F_{2}/F_{1}) = n, then for v′ ∈ V_{b} generic in V over L, we can find {b_{1}, …, b_{i}, … b_{n}} ∈ V_{b} with F_{2}(uv′b_{i}) holding. For each b_{i} we claim that there exists a unique c_{i} ∈ V_{c} such that F_{3}(uv′b_{i}c_{i}) holds. For the existence, we can use Lemma 2.7, with the simple modification that, with the notation there, if L_{m} is the algebraic closure of the field generated by ḡ(uv), then provided dim(V) ≥ 1, we can find v′ ∈ V_{v} ∩ V generic over L with uv′ ∈ L_{s}[[t^{1/r}]] for some algebraically closed field L_{s} containing L_{m}. For the uniqueness, we can use the fact that Zariski multiplicity is summable over specialisation (Lemma 2.4) and the fact that for generic (u′v′b′_{i}) ∈ V_{uvb} ∩ F_{2}, we can find a unique c′_{i} ∈ V_{c} such that F_{3}(u′v′b′_{i}c′_{i}) holds. Finally, we claim that {b_{1}c_{1}, …, b_{n}c_{n}} enumerate F_{3}(uv′xy) ∩ V_{bc}. This is clear by the above proof and the fact that {b_{1}, …, b_{n}} enumerates F_{2}(uv′x) ∩ V_{b}.

□

Lemma 2.9. The following versions of the above properties hold when we consider finite equidimensional covers, possibly with components, with the definition of Zariski multiplicity given in Definition 2.1.

Proof. For Lemma 2.3, we replace the hypotheses with F_{1} is smooth irreducible, F_{2} is an equidimensional finite cover of F_{1} and F_{3} is an etale cover of F_{2}. We then claim, using notation as in Lemma 2.2, that mult_{abc}(F_{3}/F_{1}) = mult_{ab}(F_{2}/F_{1}). By definition mult_{abc}(F_{3}/F_{1}) = ∑_{(abc)∈C_{i}}(mult_{abc}(C_{i}/F_{1})), where C_{i} are the irreducible components of F_{3} passing through (abc). As F_{3} is an etale cover of F_{2}, the images of the C_{i} are precisely the irreducible components D_{i} of F_{2} passing through (ab), each C_{i} is an etale cover of D_{i} and mult_{ab}(F_{2}/F_{1}) = ∑_{(ab)∈D_{i}}(mult_{ab}(D_{i}/F_{1})). Hence, it is sufficient to prove the result in the case when F_{2} and F_{3} are irreducible. This is just Lemma 2.3

For Lemma 2.4, we replace the hypothesis with F is an equidimensional finite cover of D. The proof then goes through exactly as in the lemma with the observation that if we find a_{n+1} ∈ V_{b} and F (a′a_{n+1}) then we can find an irreducible component C passing through (a′a_{n+1}) which allows us to apply Theorem 3.3 in [6] to obtain c with C(a′′c) and (a′′c) specialising to (a′a_{n+1}).

For Definition 2.5, we alter the hypothesis to F is an equidimensional finite cover of U × V. Again, we can use an identical proof to show that left multiplicity and right multiplicity are well defined. The proof of Lemma 2.6 with the new hypothesis on F is identical

We don’t require a modified version of Lemma 2.7, the result we need is contained in the modified proof of Lemma 2.3

For Lemma 2.8, we alter the hypotheses to F_{2} is an equidimensional cover of F_{1} and F_{3} is an etale cover of F_{2}. We then claim that for (uvb) a non-singular point of F_{2} and (uvbc) ∈ F_{3}, necessarily nonsingular as well, that RightMult_{uvbc}(F_{3}/F_{1}) = RightMult_{uvb}(F_{2}/F_{1}) and similarily for left multiplicity. To prove this, note that as (uvb) and (uvbc) are non-singular points, there exist unique components C and D passing through (uvb) and (uvbc) respectively. Now replacing C and D by the open subsets C′ and D′ of smooth points, we can apply the definition of Right Multiplicity and the proof of Lemma 2.8.

□

3. Analytic Methods

In order to use the method of etale morphisms, which preserve Zariski multiplicity, we need to work inside the Henselisation of local rings L[x_{1}, …, x_{n}]_{(x_{1},...,x_{n})}. In the next section, we will only need the result for the local ring in 2 variables L[x, y]_{(x,y)}.

We let L[[x_{1}, …, x_{n}]] denote the ring of formal power series in n variables, which is the formal completion of L[x_{1}, …, x_{n}]_{(x_{1},...,x_{n})} with respect to the canonical order valuation, see for example Section 2 of [6]. The following is a classical result, requiring the fact that etale morphisms are flat, used in the proof of the Artin approximation theorem. This relates the Henselisation of the ring L{x_{1}, …, x_{n}} of strictly convergent power series in several variables with its formal completion L[[x_{1}, …, x_{n}]], see [3] or [16];

Henselisation(L[x_{1}, … x_{n}]_{(x_{1},...x_{n})}) = L[[x_{1}, … x_{n}]] ∩ L(x_{1}, … x_{n})^{alg}

This implies that

O_{¯0,A^{n}}^{∧} ≅ L[[x_{1}, … x_{n}]] ∩ L(x_{1}, … x_{n})^{alg}

The following result, which can be found in [4], is essential for the next section

Lemma 3.1. Weierstrass Preparation

Let F (x_{1}, … x_{n}) be a polynomial in L[x_{1}, …, x_{n}] which is regular in the variable x_{n}. Then we have F (x_{1}, …, x_{n}) = U(x_{1}, …, x_{n})G(x_{1}, …, x_{n}) where U(x_{1}, …, x_{n}) is a unit in the local ring L[[x_{1}, …, x_{n}]] and G(x_{1}, …, … x_{n}) is a Weierstrass polynomial in x_{n} with coefficients in L[[x_{1}, …, x_{n−1}]]

We will require the Weierstrass decomposition to hold inside Henselisation(L[x_{1}, …, x_{n}]), therefore we need to show that the Weierstass data can be found inside L(x_{1}, …, x_{n})^{alg}. This is acheived by the following lemma;

Lemma 3.2. Definability of Weierstrass data

Let F (x_{1}, …, x_{n}) be a polynomial with coefficients in L such that F is regular in x_{n}, then if F (x_{1}, …, x_{n}) = U(x_{1}, …, x_{n})G(x_{1}, …, x_{n}) is the Weierstrass decomposition of F with G(x_{1}, …, x_{n}) = x_{n}^{m} + a_{1}(x_{1}, …, x_{n−1})x_{n}^{m−1}+…+a_{m}(x_{1}, …, x_{n−1}), and a_{i} ∈ L[[x_{1}, …, x_{n−1}]], U(x_{1}, …, x_{n}) ∈ L[[x_{1}, …, x_{n}]], then a_{i}(x_{1}, …, x_{n−1}) ∈ L(x_{1}, …, x_{n−1})^{alg} and U(x_{1}, …, x_{n}) ∈ L(x_{1}, …, x_{n})^{alg}.

Proof. This can be proved by rigid analytic methods. Equip L with a complete non-trivial non-archimedean valuation v and corresponding norm ||.||_{v}, this can be done for example by assuming that L is the completion of an algebraically closed field with any non-archimidean valuation, see [4]. Let T_{n−1}(L) be the free Tate algebra in the indeterminate variables x_{1}, …, x_{n−1} over L, that is the subalgebra of strictly convergent power series in L[[x_{1}, …, x_{n−1}]]. By the proof of Weierstrass preparation in [4], as F ∈ T_{n−1}(L)[x_{n}], the coefficients a_{i} lie in T_{n−1}(L) and U(x_{1}, …, x_{n}) ∈ T_{n−1}(L)[x_{n}]. Now choose (u_{1}, … u_{n−1}) ⊂ L transcendental over the coefficients of F with max({||u_{i}||}) ≤ 1. Then if s_{1}(ū), …, s_{m}(ū) denote the roots of F (ū, x_{n}) with ||s_{i}(ū)|| ≤ 1, then both U(ū, s_{i}(ū)) and G(ū, s_{i}(ū)) define elements of L and moreover, by a theorem in [16], we have that the coefficients a_{i}(ū) are symmetric functions of the s_{i}(ū). Hence the a_{i}(ū) belong to L(ū)^{alg}. As ū was transcendental, we have that each a_{i} ∈ L[x_{1}, …, x_{n−1}]^{alg}. As U(x_{1}, … x_{n}) = F/G(x_{1}, …, x_{n}), we clearly have that U(x_{1}, …, x_{n}) ∈ L[x_{1}, …, x_{n}]^{alg} as well.

□

4. Families of Curves in P^{2}(L)

We consider the family Q_{d} of projective curves in P^{2}(L) with degree d. An element of Q_{d} may be written;

∑_{0≤i+j≤d} a_{ij}(X/Z)^{i}(Y/Z)^{j} = 0

which, rewriting in homogenous form, becomes;

∑_{0≤i+j≤d} a_{ij}X^{i}Y^{j}Z^{d−(i+j)} = 0

For ease of notation, we will use affine coordinates x = X/Z and y = Y/Z. More generally, if we give an affine cover, we implicitly assume that it can be projectivized by taking ȳ = (y_{1}, …, y_{n}) = (Y_{1}/Z, …, Y_{n}/Z). As the notion of Zariski multiplicity is local, this will not effect our calculations.

Now consider two such families Q_{d} and Q_{e}. Then we have the cover obtained by intersecting degree d and degree e curves

Spec(L[x, y, u_{ij}, v_{ij}]/ < s(u_{ij}, x, y), t(v_{ij}, x, y) >) → Spec(L[u_{ij}, v_{ij}]). (*)

where

s(u_{ij}, x, y) = ∑_{0≤i+j≤d} u_{ij}x^{i}y^{j}

t(v_{ij}, x, y) = ∑_{0≤i+j≤e} v_{ij}x^{i}y^{j}

We denote the parameter space for degree d curves by U and the parameter space for degree e curves by V. These are irreducible affine spaces of dimension (d+1)(d+2)/2 and (e+1)(e+2)/2 respectively. The total space Q_{d} is equidimensional, as it is defined by a single polynomial s, see, for example, [10]. It is, therefore, irreducible. For suppose that Q_{d} = W_{1} ∪ W_{2}, with W_{1} and W_{2} having equal dimension, then U = pr(W_{1}) = pr(W_{2}), as U is irreducible, and W_{1}(ū) = W_{2}(ū), for generic ū ∈ U, as the corresponding curve Cū is irreducible. The condition that the fibres W_{1}(ū) and W_{2}(ū) are equal, defines a constructible subset of U, it follows that W_{1} and W_{2} intersect in an open subset of the same dimension. Hence, W_{1} = W_{2}. By the same argument, Q_{e} is irreducible. The cover (*) is generically finite, that is there exists an open subset U′ ⊂ Sp(L[u_{ij}, v_{ij}]) for which the restricted cover has finite fibres. Throughout this section, we will denote the base space of the cover by U × V, bearing in mind that we implicitly mean by this (U × V) ∩ U′. Now, given 2 fixed parameters sets ū and v̄, with (ū, v̄) ∈ U′, corresponding to curves Cū and Cv̄, the algebraic multiplicity of the cover (∗) at (00, ū, v̄) is exactly the intersection multiplicity I(Cū, Cv̄, 00) of the curves at (00). The cover (*) is equidimensional as U × V satisfies the presmoothness axiom with the smooth projective variety P^{2}(L). Restricting to a finite cover over U′, by definition 2.1 we can also define the Zariski multiplicity of the cover at the point (00, ū, v̄). The main result that we shall prove in this paper is the following, which generalises an observation given in [12];

Theorem 4.1. In all characteristics, the algebraic multiplicity and Zariski multiplicity of the cover (∗) coincide at (00, ū, v̄).

Definition 4.2. We say that a monic polynomial p(x, ȳ) is Weierstrass in x if p(x, ȳ) = x^{n} + … + q_{j}(ȳ)x^{n−j} + … + q_{n}(ȳ) with q_{j}(¯0) = 0.

Definition 4.3. Let F (x, ȳ) be a polynomial in x with coefficients in L[ȳ]. We say the cover

Spec(L[xȳ]/ < F (x, ȳ) >) → Spec(L[ȳ])

is generically reduced if for generic ū ∈ Spec(L[ȳ]), F (x, ū) has no repeated roots.

Definition 4.4. Let F → U ×V be a finite cover with U and V smooth, such that for (ū, v̄) ∈ U ×V the fibre F (ū, v̄) consists of the intersection of algebraic curves Fū, Fv̄. We call the family sufficiently deformable at (ū_{0}, v̄_{0}) if there exists ū′ ∈ U generic over L such that F_{}_{ū′} intersects Fv̄_{0} transversely at simple points.

We now require a series of lemmas;

Lemma 4.5. Let F (x, ȳ) be a Weierstrass polynomial in x with F (0, ¯0) = 0 then algebraic multiplicity and Zariski multiplicity coincide at (0, ¯0) if the cover

Spec(L[xȳ]/ < F (x, ȳ) >) → Spec(L[ȳ])

is generically reduced.

Proof. We have that F (x, ȳ) = x^{n} + q_{1}(ȳ)x^{n−1} + … + q_{n}(ȳ) where q_{i}(¯0) = 0. The algebraic multiplicity is given by length(L[x]/F (x, ¯0)) = ord(F (x, ¯0) = n in the ring L[x] with the canonical valuation. We first claim that the Zariski multiplicity is the number of solutions to x^{n}+q_{1}(ϵ̄)x^{n−1}+…+q_{n}(ϵ̄) = 0 (†), where ϵ̄ is generic in V¯0. For suppose that (a, ϵ̄) is such a solution, then F (a, ϵ̄) = 0 and by specialisation F (π(a), ¯0) = 0. As F is a Weierstrass polynomial in x, π(a) = 0, hence a ∈ V_{0}, giving the claim. We have that Disc(F (x, ȳ)) = Resȳ(F, \frac{∂F}{∂x}) is a regular polynomial in ȳ defined over L. By the fact that the cover is generically reduced, this defines a proper closed subset of Spec(L[ȳ]). Therefore, Disc(F (x, ȳ))|ϵ̄ ≠ 0, hence (†) has no repeated roots. This gives the lemma. □

Lemma 4.6. Let F (x, ȳ) be any polynomial with F (x, ¯0) ≠ 0 and F (0, ¯0) = 0. Then if the cover Spec(L[x, ȳ]/ < F (x, ȳ) >) → Spec(L[ȳ]) is generically reduced, the Zariski multiplicity at (0, ¯0) equals ord(F (x, ¯0)) in L[x].

Proof. By the Weierstrass Preparation Theorem, Lemma 3.1, we can write F (x, ȳ) = U(x, ȳ)G(x, ȳ) with U(x, ȳ), G(x, ȳ) ∈ L[[x, ȳ]], G(x, ȳ) a Weierstrass polynomial in x and deg(G) = ord(F (x, ¯0)), see also the more closely related statement given in [2]. By Lemma 3.2, we may take the new coefficients to lie inside the Henselized ring L[x, ȳ]∧ ; ¯0, hence inside some finite etale extension L[x, ȳ]^{ext} of L[x, ȳ] (possibly after localising L[x, ȳ] corresponding to an open subset of Spec(L[x, ȳ]) containing (0, ¯0)). Now we have the sequence of morphisms;

Sp(L[x, ȳ]^{ext}/UG) → Spec(L[x, ȳ]/F) → Spec(L[ȳ])

The left hand morphism is etale at ¯0, hence by Lemma 2.3 or Lemma 2.7, to compute the Zariski multiplicity of the right hand morphism, we need to compute the Zariski multiplicity of the cover

Spec(L[x, ȳ]^{ext}/UG) → Spec(L[ȳ])

at (0, ¯0)^{lift}, the marked point in the cover above (0, ¯0). Choose ϵ̄ ∈ V¯0, the fibre of the cover is given formally analytically by L[[x, ȳ]]/ < UG > ⊗_{L[ȳ],ȳ↦ϵ̄}L, hence by solutions to U(x, ϵ̄)G(x, ϵ). By definition of Zariski multiplicity, we consider only solutions (xϵ̄) in V_{(0,¯0)^{lift}}. As U(x, ȳ) is a unit in the local ring L[x, ȳ]_{(0,¯0)^{lift}}^{ext}, we must have U(x, ϵ̄) ≠ 0 for such solutions, otherwise by specialisation U((0, ¯0)^{lift}) = 0. Hence, the solutions are given by G(x, ϵ̄) = 0. Now, we use the previous lemma to give that the Zariski multiplicity is exactly deg(G) as required.

□

Now return to the cover

Sp(L[x, y, u_{ij}, v_{ij}]/ < s(u_{ij}, x, y), t(v_{ij}, x, y) >) → Sp(L[u_{ij}, v_{ij}]) (*)

We will show below, Lemma 4.12, that this is a sufficiently deformable family at (ū_{0}, v̄_{0}) when Cū_{0} and Cv̄_{0} define reduced curves. We claim the following;

Lemma 4.7. Suppose parameters ū^{0} and v̄^{0} are chosen such that C_{}_{ū^{0}} and C_{}_{v̄^{0}} are reduced Weierstrass polynomials in x. Then the Zariski multiplicity of the cover (∗) at (0, 0, ū^{0}, v̄^{0}) equals the intersection multiplicity I(C_{}_{ū^{0}}, C_{}_{v̄^{0}}, (0, 0)) of C_{}_{ū^{0}} and C_{}_{v̄^{0}} at (0, 0).

Proof. Introduce new parameters ū′ and v̄′. Let C\begin{matrix} & ū′ \\ & ū^{0} \end{matrix} and C\begin{matrix} & v̄′ \\ & v̄^{0} \end{matrix} denote the curves C_{}_{ū^{0}} and C_{}_{v̄^{0}} deformed by the parameters ū′ and v̄′ respectively. That is C\begin{matrix} & ū′ \\ & ū^{0} \end{matrix} is given by the new equation Σ_{1≤i+j≤d}(u_{ij}^{0} + u′_{ij})x^{i}y^{j}. Let

F (y, ū′, v̄′) = Res(C\begin{matrix} & ū′ \\ & ū^{0} \end{matrix}, C\begin{matrix} & v̄′ \\ & v̄^{0} \end{matrix}). Then,

F (0, ¯0, ¯0) = Res(s(u_{ij}^{0}, x, 0), t(v_{ij}^{0}, x, 0)) = 0

as C_{}_{ū^{0}} and C_{}_{v̄^{0}} are Weierstrass in x and share a common solution at (0, 0). By a result due to Abhyankar, see for example [1], ord_{y}(F (y, ¯0, ¯0)) = Σ_{x}I(C_{}_{ū^{0}}, C_{}_{v̄^{0}}, (x0)) at common solutions (x, 0) to C_{}_{ū^{0}} and C_{}_{v̄^{0}} over y = 0. As C_{}_{ū^{0}} and C_{}_{v̄^{0}} are Weierstrass polynomials in x, this is just I(C_{}_{ū^{0}}, C_{}_{v̄^{0}}, (0, 0)). By the previous lemma and the fact that F (y, ū, v̄) is generically reduced (see argument (†) below), it is therefore sufficient to prove that the Zariski multiplicity of the cover (∗) at (00, ū^{0}, v̄^{0}) equals the Zariski multiplicity of the cover Spec(L[y, ū′, v̄′]/ < F (y, ū′, v̄′) >) → Spec(L[ū′, v̄′]) (**) at (0, ¯0, ¯0). Suppose the Zariski multiplity of (∗∗) equals n. Then there exist distinct y_{1}, …, y_{n} ∈ V_{0} and (δ̄, ϵ̄) generic in V_{(¯0,¯0)} ∩ U × V such that F (y_{i}, δ̄, ϵ̄) holds. Consider Q(ū′, v̄′) = Res(F (y, ū′, v̄′), ∂F/∂y(y, ū′, v̄′)). This defines a closed subset of U × V defined over L, we claim that this in fact proper closed (†). By the fact that the family is sufficently deformable at (ū_{0}, v̄_{0}), we can find (ū, v̄_{0}) such that Cū intersects Cv̄_{0} transversely at simple points. Without loss of generality, making a linear change of coordinates, we may suppose that for there do not exists points of intersection of the form (x_{1}y) and (x_{2}y) for x_{1} ≠ x_{2}. By Abhyankar’s result, this implies that F (y, ū′, v̄_{0}) has no repeated roots. Then, by genericity of (δ̄, ϵ̄), we have that Q(δ̄, ϵ̄) ≠ 0. Hence F (y_{i}, δ̄, ϵ̄) is a non-repeated root. By Abhyankar’s result, we can find a unique x_{i} with (x_{i}y_{i}) a common solution to the deformed curves Cδ̄ ; ū^{0} and Cϵ̄ ; v̄^{0}. We claim that each (x_{i}y_{i}) ∈ V_{00}. As Cδ̄ ; ū^{0}(x_{i}y_{i}) = 0, by the fact (ū^{0}, δ̄, y_{i}) specialises to (ū^{0}, ¯0, 0) and C_{}_{ū^{0}} is a Weierstrass polynomial in x, we have that π(x_{i}) = 0 as well. This shows that the Zariski multiplicity of the cover (∗) is at least n. A virtually identical argument shows that the Zariski multiplicity of the cover (∗) is at most n as well.

□

We now have the following result;

Lemma 4.8. Let C_{}_{ū^{0}} and C_{}_{v̄^{0}} be reduced curves, having finite intersection, then the Zariski multiplicity of the cover (∗) at ((0, 0), ū^{0}, v̄^{0}) equals the intersection multiplicity I(C_{}_{ū^{0}}, C_{}_{v̄^{0}}, (0, 0)) of C_{}_{ū^{0}} and C_{}_{v̄^{0}} at (0, 0).

Proof. We have C_{}_{ū^{0}} = s(u_{ij}^{0}, x, y) and C_{}_{v̄^{0}} = t(v_{ij}^{0}, x, y). By making the substitutions Ū = ū^{0} + ū and V̄ = v̄^{0} + v̄, we may assume that ū^{0} = v̄^{0} = ¯0. Moreover, we can suppose that;

s(¯0_{ij}, x, 0) ≠ 0 and t(¯0_{ij}, x, 0) ≠ 0. (**)

This can be achieved by making the invertible linear change of variables (x′ = x, y′ = λx + μy) with (λ, μ) ∈ L^{2} and μ ≠ 0, noting that as Cū_{0} and Cv̄_{0} are curves, for some choice of (λ, μ), the corresponding polynomials s(u_{ij}^{0}, x, y) and t(v_{ij}^{0}, x, y) do not vanish identically on the line λx + μy = 0. It is trivial to check that the transformation preserves both Zariski multiplicity and intersection multiplicity, so our calculations are not effected.

We may then apply the Weierstrass preparation theorem, Lemma 3.1, in the ring L[[u_{ij}, v_{ij}, x, y]], obtaining factorisations s(u_{ij}, x, y) = U_{1}(u_{ij}, x, y)S(u_{ij}, x, y) and t(v_{ij}, x, y) = U_{2}(v_{ij}, x, y)T (v_{ij}, x, y) where U_{1} and U_{2} are units in the local rings L[[u_{ij}, x, y]] and L[[v_{ij}, x, y]], S, T are Weierstrass polynomials in x with coefficients in L[[u_{ij}, y]] and L[[v_{ij}, y]] respectively. A close inspection of the Weierstrass preparation theorem, see [2], shows that we can obtain the following uniformity in the parameters ū and v̄;

Namely, if U = {u_{ij} : s(u_{ij}, x, 0) ≠ 0} and V = {v_{ij} : t(v_{ij}, x, 0) ≠ 0}, are the constructible sets for which (∗) holds, then if we let R_{U} and R_{V} denote the coordinate rings of U and V, we may assume U_{1}, U_{2} lie in R_{U}[[x, y]] and the coefficients of S, T lie in R_{U}[[y]] and R_{V} [[y]] respectively. By Lemma 3.2, we may assume that U_{1}, U_{2}, S and T lie in a finite etale extension R_{U×V} [x, y]^{ext} of the algebra A = R_{U×V} [x, y] (again, possibly after localisation corresponding to an open subvariety of Spec(A). Now we have the sequence of morphisms;

Spec(\frac{R_{U×V} [x,y]^{ext}}{<U_{1}S,U_{2}T >}) → Spec(\frac{R_{U×V} [x,y]}{<s,t>}) → Spec(R_{U×V}).

We claim that the left hand morphism is etale at the point (¯0, ¯0, (00)^{lift}). This follows from the fact that R_{U×V} [x, y]^{ext} is an etale extension of R_{U×V} [x, y] and the maximal ideal given by (¯0, ¯0, (00)^{lift}) contains < U_{1}S, U_{2}T >. Now consider the cover;

Spec(\frac{R_{U×V} [x,y]^{ext}}{<U_{1}S,U_{2}T >}) → Spec(R_{U×V}) (***)

For ū, v̄ in U × V, the fibre of this cover over ū, v̄ corresponds exactly to the intersection of the reducible curves C′ ; ū and C′ ; v̄ which lift the original curves Cū and Cv̄ to an etale cover of Spec(L[xy]). By Theorem 1.8 and Lemma 2.3 (in the case when Cū_{0}, Cv̄_{0} intersect at simple points) or Lemma 2.7 (for singular points of intersection) and the corresponding Lemma 2.9 for reducible covers, it is sufficent to show that the Zariski multiplicity of the cover (***) at (¯0, ¯0, (00)^{lift}) corresponds to the intersection multiplicity of the curves C′ ; ū_{0}, C′ ; v̄_{0} at (00)^{lift}. The idea now is to apply Lemma 4.7 to the Weierstrass factors of C′ ; ū and C′ ; v̄. This will be achieved by the ”unit removal” lemma below (Lemma 4.15).

□

In order to prove the ”unit removal lemma”, we first require some more definitions and a moving lemma for curves;

Definition 4.9. Let X → Spec(L[x, y]) be an etale cover in a neighboorhood of (0, 0), with distiguished point (0, 0)^{lift}. We call a curve C on X passing through (0, 0)^{lift} Weierstrass if, in the power series ring L[[x, y]], the defining equation of C may be written as a Weierstrass polynomial in x with coefficients in L[[y]]. Definition 4.10. Let F → U × V be a finite equidimensional cover of a smooth base of parameters U × V with a section s : U × V → F. We call the cover Weierstrass with units if the fibres F (ū, v̄) can be written as the intersection of reducible curves C′ ; ū and C′ ; v̄ in an etale cover A_{ū,v̄} of U_{ū,v̄} ⊂ Spec(L[x, y]) with the distinguished point s(ū, v̄) lying above (0, 0) and C′ ; ū, C′ ; v̄ factoring as UūFū and Uv̄Fv̄ with Uū, Uv̄ units in the local ring O_{s(ū,v̄),A_{ū,v̄}} and Fū, Fv̄ Weierstrass curves in A_{ū,v̄}.

Let hypotheses on F, U and V be as above. We call the cover Weierstrass if the fibres F (ū, v̄) can be written as above but with C′ ; ū, C′ ; v̄ Weierstrass curves in A_{ū,v̄}.

We say that a Weierstrass cover (with units) factors through the family of projective degree d and degree e curves if the cover F → U × V factors as F → F′ → U ×V where F′ → U ×V is the finite equidimensional cover obtained by intersecting the families Q_{d} and Q_{e} restricted to U and V.

Lemma 4.11. The cover (***) in Lemma 4.8 is a Weierstrass cover with units factoring through the family of projective degree d and degree e curves. Proof. Clear by the above definitions.

□

Lemma 4.12. Moving Lemma for Reduced Curves

Let Q_{d} and Q_{e} be the families of all projective degree d and degree e curves. That is, with the usual coordinate convention x = X/Z, y = Y/Z, Q_{d} consists of all curves of the form s(ū, x, y) = ∑_{0≤i+j≤d} u_{ij}x^{i}y^{j}. Then, if ū, v̄ are chosen in L, so that the reduced curves Cū and Cv̄ are defined over L, if the tuple ū′ is chosen to be generic in U over L, the deformed curve Cū′ ; ū intersects Cv̄ transversely at simple points. Proof. We can give an explicit calculation;

Let Cū′ ; ū be defined by the equation s(ū′, x, y) = ∑_{0≤i+j≤d} u′_{ij}x^{i}y^{j}

and Cv̄ by t(v̄, x, y) = ∑_{0≤i+j≤e} v_{ij}x^{i}y^{j} with {v_{ij} : 0 ≤ i + j ≤ e} ⊂ L and {u′_{ij} : 0 ≤ i + j ≤ d} algebraically independent over L. Let (x_{0}y_{0}) be a point of intersection, then dim(x_{0}y_{0}/L) = 1, otherwise dim(x_{0}y_{0}/L) = 0 and, as L is algebraically closed, we must have that x_{0}, y_{0} ∈ L. Substituting (x_{0}y_{0}) into the equation s(ū′, x, y) = 0, we get a non trivial linear dependence over L between u′_{00} and u′_{ij} for 1 ≤ i + j ≤ d which is impossible. Now, the locus of singular points for Cv̄ is defined over L and hence (x_{0}y_{0}) is a simple point of Cv̄. Now we further claim that s(ū′, x, y) = 0 defines a non-singular curve in P^{2}(K_{ω}) with transverse intersection to Cv̄ Consider the conditions Sing(ū) given by ∃x_{0}∃y_{0}((\frac{∂s}{∂x}(x_{0}y_{0}), \frac{∂s}{∂y}(x_{0}y_{0})) = (0, 0)) and Non-Transverse(ū) by ∃x_{0}∃y_{0}(\frac{∂s}{∂x}(x_{0}y_{0})\frac{∂t}{∂y}(x_{0}y_{0}) − \frac{∂s}{∂y}(x_{0}y_{0})\frac{∂t}{∂x}(x_{0}y_{0}) = 0) By the properness of P^{2}(K_{ω}), these conditions define closed subsets of the parameter space U defined over L. We claim that this in fact a proper closed subset. This can be proved in a number of ways. In the case where we restrict ourselves to affine curves, the result follows from a classical result of Kleiman, see [9], as affine space A^{2}(K_{ω}) is homogenous for the action of the additive group (A^{2}(K_{ω}), +). More generally, we can use the moving lemma, given in [8], by observing that the class of all degree d projective curves is closed under rational equivalence. We can also give an explicit proof using Bertini’s theorem;

Observe that the curve Cū defines a complete linear system |Cū| corresponding exactly to the zero loci of sections σ of the bundle O_{P^{2}}(d). We claim the following;

(i). The system |Cū| is base point free.

(ii). The system |Cū| separates points.

(i) is obvious. Just observe that if F (x, y) defines a polynomial of degree d, then we can form a pencil of projective degree d curves G(X, Y, Z) + λZ^{d} such that F (X/Z, Y/Z) = G(X, Y, Z)/Z^{d} where x = X/Z, y = Y/Z. Clearly, for λ_{1} ≠ λ_{2}, the curves F (x, y) = λ_{1} and F (x, y) = λ_{2} are disjoint.

(ii) is also clear. Given distinct points p and q in P^{2}(K), we can find a line l passing through p but not q. Then the d-fold line l^{d} belongs to the system |Cū|.

Now we can define a morphism Φ_{d} : P^{2}(K) → P^{d(d+3)/2}(K), by sending x ∈ P^{2} to the hyperplane H_{x} ⊂ U of curves of degree d, passing through x. By (i) and (ii), the restriction of Φ_{d} to Cv̄ is injective. By arguments on Frobenius for curves, given in [6], we can assume that Φ_{d} is an immersion. Using Bertini’s Theorem, a generic hyperplane H_{}_{ū′} of P^{d(d+3)/2}(K) will intersect Im(Cv̄) transversely in simple points. By definition of the morphism Φ_{d}, and the fact that it is an immersion, the corresponding curve Cū′ ; ū also intersects Cv̄ transversely in simple points.

One can also give an enumerative calculation, which was done in an older version of this paper, see [5], but it seems unnecessary.

□

Remarks 4.13. If we restrict the family of curves, the result in general fails. A simple example is given by the family of all projective degree 3 curves Q_{3}^{0,0} passing through (0, 0) with x = X/Z and y = Y/Z. If we take Cv̄ to be the cusp x^{2} − y^{3}, then any curve in Q_{3}^{0,0} will have a non-transverse intersection with Cv̄ at the origin.

Lemma 4.14. Moving Lemma for Curves with Finitely Many Marked Points

Let hypotheses be as in the previous lemma with Cū and Cv̄ defining reduced curves. Suppose also that there exists finitely many marked points {p_{1}, …, p_{n}} on Cv̄ defined over L. Then for ū′ ∈ U generic over L the deformed curve Cū′ ; ū intersects Cv̄ transversely at finitely many simple points excluding the set {p_{1}, …, p_{n}}.

Proof. As before, the condition that ū′ defines a curve Cū′ ; ū either with non-transverse intersection to Cv̄ or passing through at least one of the points {p_{1}, …, p_{n}} is a closed subset of U defined over L. Using the above proof and the obvious fact that we can find a curve Cū′ ; ū not passing through any of the points {p_{1}, …, p_{n}}, we see that it is proper closed.

□

Lemma 4.15. Unit Removal for Reduced Curves

Let (π, s) : F → U × V be a Weierstrass cover with units factoring through projective degree d and degree e curves. Let (ū, v̄) ∈ U ×V, then there exists a Weierstrass cover (π′, s′) : F^{−} → U′×V′ with U′ ⊂ U and V′ ⊂ V open subsets, (ū, v̄) ∈ U′ × V′, such that Mult_{(ū,v̄,s(ū,v̄))}(F/U × V) = Mult_{ū,v̄,s′(ū,v̄))}(F^{−}/U′ × V′).

Proof. Let C′ ; ū and C′ ; v̄ be the Weierstrass curves with units in A_{ū,v̄} lifting the curves Cū and Cv̄. Now suppose that Mult_{ū,v̄,s(ū,v̄)}(F/U × V) = n. Then we can find (ū′, v̄′) ∈ V_{ūv̄} ∩ U × V generic over L such that the deformed curve Cū′ ; ū intersects Cv̄′ ; v̄ at the n distinct points x_{1}, …, x_{n} in V_{s(ū,v̄)}. Now using the Weierstrass factorisations of Cū′ ; ū and Cv̄′ ; v̄, we claim that U ū′ ; ū(x_{i}) ≠ 0 and U v̄′ ; v̄(x_{i}) ≠ 0. Suppose not, then U ū′ ; ū(x_{i}) = U v̄′ ; v̄(x_{i}) = 0 and as (ū′, v̄′, x_{i}) specialises to (ū, v̄, s(ū, v̄)), then Uū(s(ū, v̄)) = Uv̄(s(ū, v̄)) = 0. This contradicts the fact that Uū and Uv̄ are units in the local ring O_{s(ū,v̄),A_{ū,v̄}}. Therefore, we must have

that F ū′ ; ū(x_{i}) = F v̄′ ; v̄(x_{i}) = 0. This shows that Mult_{ū,v̄,s(ū,v̄)}(F^{−}/U ×

V) ≥ n where F^{−} → U × V is the cover of U × V obtain by taking as fibres F^{−}(ū, v̄) the intersection of the Weierstrass factors Fū and Fv̄. Formally, if F is defined by Spec(\frac{R_{U×V} [x,y]^{ext}}{<U_{1}S,U_{2}T >}) then F^{−} is defined by

Spec(\frac{R_{U×V} [x,y]^{ext}}{<S,T >}). Clearly as F^{−} ⊂ F is a union of components of F, we have that Mult_{ū,v̄,s(ū,v̄)}(F^{−}/U × V) ≤ n as well. This proves the lemma.

□

We now complete the proof of Lemma 4.8. By unit removal, it is sufficient to compute the Zariski multiplicity of the cover

Spec(\frac{R_{U×V} [x,y]^{ext}}{<S,T >}) → Spec(R_{U×V})

The fibre over (ū, v̄) of this cover corresponds exactly to the intersection of the Weierstrass curves Fū and Fv̄ lifting Cū and Cv̄. We then use Lemma 2.7, noting that the Weierstrass factors are still reduced, see [2], to finish the result, with the straightforward modification that we work in a uniform family of etale covers.

We now turn to the problem of non-reduced curves. We will show the following stronger version of Lemma 4.8

Lemma 4.16. Let C_{}_{ū^{0}} and C_{}_{v̄^{0}} be non-reduced curves having finite intersection, then the Zariski multiplicity of the cover (*) at ((0, 0), ū^{0}, v̄^{0}) equals the intersection multiplicity I(C_{}_{ū^{0}}, C_{}_{v̄^{0}}, (0, 0)) of C_{}_{ū^{0}} and C_{}_{v̄^{0}} at (0, 0).

First, we will require some more lemmas.

Lemma 4.17. Let Cū_{0} and Cv̄_{0} be reduced curves intersecting transversely at (0, 0). Then the Zariski multiplicity, left multiplicity and right multiplicity of the cover (*) at ((0, 0), ū^{0}, v̄^{0}) equals 1.

Proof. First note that by Lemma 2.6 (and corresponding Lemma 2.9), and the fact that a generic deformation Cv̄′ ; v̄_{0} will still intersect Cū_{0} transversely by Lemma 4.12, it is sufficient to prove the result for right multiplicity.

In order to show this we require the following result, given for analytic curves in [2], we will only need the result for polynomials;

Implicit Function Theorem:

If G(X, Y) is a power series with G(0, 0) = 0 then G_{Y} (0, 0) ≠ 0 implies there exists a power series η(X) with η(0) = 0 such that G(X, η(X)) = 0.

In order to show that RightMult_{(0,0),ū^{0},v̄^{0}}(F′/U × V) = 1, where F′ is the family obtained by intersecting degree d and degree e curves, we apply the implicit function theorem to the curve C_{}_{ū^{0}} at the point (0, 0) of intersection with C_{}_{v̄^{0}}. Let G(X, Y) and H(X, Y) denote the polynomials defining the curves. We have that G(0, 0) = H(0, 0) = 0. Moreover, as the first curve is non-singular at (0, 0), we may also assume that G_{Y} (0, 0) ≠ 0. Now let η(X) be given by the theorem. As the intersection of the curves C_{}_{ū^{0}} and C_{}_{v̄^{0}} is transverse, ord_{X}H(X, η(X)) = 1.Now we have the sequence of maps;

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

where L[X]^{ext} is an etale extension of L[X] containing η(X). (Note that η(X) is trivially algebraic over L(X)). This corresponds to a sequence of finite covers F_{1} → F′(u_{0}, V) → Spec(L[v̄]). The left hand morphism is trivially etale at (v̄^{0}, (00)^{lift}), hence it is sufficient to compute the Zariski multiplicity of F′ → Spec(L[v̄]) at (v̄^{0}, (00)^{lift}) by Lemma 2.3 (or corresponding Lemma 2.9). This is a straightforward calculation, the fibre over v̄^{0} consists of the scheme Spec(\frac{L[X,η(X)]}{G(X,η(X))}) = Spec(L) as ord_{X}(H(X, η(X))) = 1, hence is etale at the point (v̄^{0}, (00)^{lift}). By Theorem 1.4, the Zariski multiplicity is 1.

□

Lemma 4.18. Let hypotheses be as in Lemma 4.17, then for any (ū′, v̄′) ∈ V_{(ū^{0},v̄^{0})}, we have that Card(F′(ū′, v̄′) ∩ V_{0,0}) = 1 Proof. This follows immediately from Lemma 4.17 and Lemma 2.4.

□ Definition 4.19. For ease of notation, given curves Cū and Cv̄ of degree d and degree e intersecting at x ∈ P^{2}(K_{ω}), we define Mult_{x}(Cū, Cv̄) to be the corresponding Zariski multiplicity of the cover F′ → U × V at the point (x, ū, v̄). Similarly for left/right multiplicity.

We can now give the proof of Lemma 4.16;

Proof. Case 1. Cv̄_{0} is a reduced curve (possibly having components). Write C_{}_{ū^{0}} as G_{1}^{n_{1}}(X, Y) … G_{m}^{n_{m}}(X, Y) = 0 with G_{i} the reduced irreducible components of Cū_{0} with degree d_{i} passing through (0, 0). Choose ϵ̄_{1}^{1}, … ϵ̄_{1}^{n_{1}}, … ϵ̄_{i}^{j}, … ϵ̄_{m}^{1}, …, ϵ̄_{m}^{n_{m}} independent generic in U_{i}, the parameter space for degree d_{i} projective curves with ϵ̄_{i}^{j} ∈ V_{}_{ū_{i}^{0}}, where ū_{i}^{0} defines G_{i}. By repeated application of Lemma 4.14, the deformed curves G_{i}^{ϵ̄_{i}^{j}} = 0 intersect Cv̄_{0} transversely at disjoint sets of points We denote by Z_{}_{ϵ̄_{i}^{j}} those points lying in V_{00}. Now the curve defined by

∏_{ij} G_{i}^{ϵ̄_{i}^{j}} = 0 is a deformation Cϵ̄ ; ū^{0} of C_{}_{ū^{0}}. We let Zϵ̄ denote the points of intersection of Cϵ̄ ; ū^{0} with Cv̄_{0} in V_{00}. Then we have;

Zϵ̄ = ⋃_{ij} Z_{}_{ϵ̄_{i}^{j}}

Card(Zϵ̄) = ∑_{ij} Card(Z_{}_{ϵ̄_{i}^{j}})

By Lemma 2.4, we have that

LeftMult_{(00)}(C_{}_{ū^{0}}, C_{}_{v̄^{0}}) = ∑_{x∈Zϵ̄} LeftMult_{x}(Cϵ̄ ; ū^{0}, C_{}_{v̄^{0}})

= ∑_{i,j} ∑_{x∈Z_{ϵ̄{j \atop i}}} LeftMult_{x}(Cϵ̄ ; ū^{0}, C_{}_{v̄^{0}}) (*)

We now claim that for a point x ∈ Z_{}_{ϵ̄_{i}^{j}},

LeftMult_{x}(Cϵ̄ ; ū^{0}, C_{}_{v̄^{0}}) = LeftMult_{x}(G_{i}^{ϵ̄_{i}^{j}}, Cv̄_{0}) (**)

This follows as both the reduced curves Cϵ̄ ; ū_{0} and G_{i}^{ϵ̄_{i}^{j}} intersect Cv̄_{0} transversely at x. Hence, in both cases the left multiplicity is 1, by Lemma 4.17.

Combining (∗) and (∗∗), we obtain;

LeftMult_{(00)}(C_{}_{ū^{0}}, C_{}_{v̄^{0}}) = ∑_{i,j} ∑_{x∈Z_{ϵ̄{j \atop i}}} LeftMult_{x}(G_{i}^{ϵ̄_{i}^{j}}, C_{}_{v̄^{0}})

A NON-STANDARD BEZOUT THEOREM FOR CURVES 25

Now using Lemma 2.4 again gives that; LeftMult_{(00)}(C_{}_{ū^{0}}, C_{}_{v̄^{0}}) = ∑_{i=1}^{m} n_{i}LeftMult_{(00)}(G_{i}, C_{}_{v̄^{0}}) (***) If we go through exactly the same calculation with Mult replacing

Left Mult, we see as well that Mult_{(00)}(C_{}_{ū^{0}}, C_{}_{v̄^{0}}) = ∑_{i=1}^{m} n_{i}Mult_{(00)}(G_{i}, C_{}_{v̄^{0}}) By Lemma 4.8, this gives Mult_{(00)}(C_{}_{ū^{0}}, C_{}_{v̄^{0}}) = ∑_{i=1}^{m} n_{i}I(G_{i}, C_{}_{v̄^{0}}, (00)) By a straightforward algebraic calculation, see the references below

at the end of the proof for the required more general result, this gives Mult_{(00)}(C_{}_{ū^{0}}, C_{}_{v̄^{0}}) = I(C_{}_{ū^{0}}, C_{}_{v̄^{0}}, (00)) as required. Case 2. Both Cū_{0} and Cv̄_{0} define non-reduced curves. Write Cū_{0} as

above and Cv̄_{0} as H_{1}^{e_{1}} … H_{n}^{e_{n}} with H_{i} the reduced compoments with degree c_{i} of Cv̄_{0} passing through (00). Then H_{1} … H_{n} = 0 defines a reduced curve passing through (00). Now repeat the argument in Case 1 for the curves Cū_{0} and H_{1} … H_{n} = 0. Again let Zϵ̄ be the intersection points of the deformed curve Cϵ ; ū_{0} with H_{1} … H_{n} = 0 in V_{(00)}. By (***) of Case 1, Lemma 2.4 and Lemma 4.18 with the fact that the intersection of Cϵ̄ ; ū^{0} with H_{1} … H_{n} is transverse, we have;

Card(Zϵ̄) = ∑_{i=1}^{m} n_{i}Mult_{(00)}(G_{i}, H_{1} … H_{n}) Now using the argument in Case 1 applied to the reduced curves G_{i}

and H_{1} … H_{n}, we have; Card(Zϵ̄) = ∑_{i=1}^{m} n_{i} ∑_{j=1}^{n} I(G_{i}, H_{j}, (00)) (*) We claim that for any component H_{j} Card(H_{j} ∩ Zϵ̄) = ∑_{i=1}^{m} n_{i}I(G_{i}, H_{j}, (00))

This follows as the deformed curve Cϵ̄ ; ū_{0} a fortiori intersects H_{j} transversely at simple points. Therefore, again by Case 1, gives the expected multiplicity. Now, using this together with (*), we write Zϵ̄ as ∪_{j}Zj ; ϵ̄ where Zj ; ϵ̄ are the disjoint sets consisting of the intersection of Cϵ̄ ; ū_{0} with H_{j}. Then by Lemma 2.6, we have that

Mult_{(00)}(C_{}_{ū^{0}}, C_{}_{v̄^{0}}) = ∑_{j} ∑_{x∈Zj ; ϵ̄} RightMult_{x}(Cϵ̄ ; ū^{0}, C_{}_{v̄^{0}})

We can now calculate the Right Mult term by applying Case 1 to the intersection of Cv̄_{0} with the reduced curve Cϵ̄ ; ū_{0} at the points of intersection x ∈ Zj ; ϵ̄. At a point x ∈ Zj ; ϵ̄, we have that

RightMult_{x}(Cϵ̄ ; ū^{0}, C_{}_{v̄^{0}}) = e_{j}I(Cϵ̄ ; ū^{0}, H_{j}, x) = e_{j}

as the intersection is transverse. Finally this gives;

Mult_{(00)}(C_{}_{ū^{0}}, C_{}_{v̄^{0}}) = ∑_{i=1}^{m} ∑_{j=1}^{n} n_{i}e_{j}I(G_{i}, H_{j}, (00))

By an algebraic result, see [11] for the case of complex algebraic curves, or [7] for its generalisation to algebraic curves in arbitrary characteristics, we have

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

as required.

□

The following version of Bezout’s theorem in all characteristics is now an easy generalisation from the above lemma. For curves C_{1} and C_{2} in P^{2}(L), we let M(C_{1}, C_{2}, x) denote the intersection multiplicity or the Zariski multiplicity, we know from the above that the two are equivalent.

Theorem 4.20. Non-Standard Bezout

Let C_{1} and C_{2} be projective curves of degree d and degree e in P^{2}(L), possibly with non-reduced components, intersecting at finitely many points {x_{1}, …, x_{i}, … x_{n}}, then we have;

∑_{i=1}^{n} M(C_{1}, C_{2}, x_{i}) = de.

Of course, we could just quote the algebraic result given in [9] (though this in fact only holds for reduced curves). Instead we can give a nonstandard proof, which in many ways is conceptually simpler and doesn’t involve any algebra;

Proof. Let Q_{d} and Q_{e} be the families of all projective degree d and degree e curves. Then we have the cover F → U × V with F ⊂ U × V × P^{2}(L) obtained by intersecting the families Q_{d} and Q_{e}. We have that

∑_{i=1}^{n} M(C_{1}, C_{2}, x_{i}) = ∑_{i=1}^{n} Mult_{x_{i}∈F (ū_{0},v̄_{0})}(F/U × V)

where (ū_{0}, v̄_{0}) define C_{1} and C_{2}. By Lemma 4.3 in [6], this equals

∑_{x∈F (ū,v̄)} Mult_{x,ū,v̄}(F/U × V)

where (ū, v̄) is generic in U × V. Using, for example, the proof of Lemma 4.12, generically independent curves Cū and Cv̄ intersect transversely at a finite number of simple points. Hence, by Lemma 4.17, the Zariski multiplicity calculated at these points is 1. As the cover F has degree de, there is a total number de of these points as required.

□

References

[1] S.S. Abhyankar, Algebraic Space Curves, Les Presses de l’Universite de Montreal, Montreal, Canada (1971)

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

[3] M. Artin, On the solutions of analytic equations. Invent. Math., 5(1968), 277-291

[4] S. Bosch, U. Guntzer, R. Remmert, Grundlehren der mathematischen Wissenscaften 261, Springer (1984)

[5] T. de Piro, A Non-Standard Bezout Theorem AG/LO Arxiv (0406176)

[6] T. de Piro, Zariski Structures and Algebraic Curves, available on my website, (2007), old version ”Zariski Structures and Algebraic Geometry” on AG arXiv (0402301) (2004)

28 TRISTRAM DE PIRO

[7] W. Fulton, Algebraic Curves, Mathematics Lecture Notes Series, Bejamin/Cummings(1969)

[8] W. Fulton, Intersection Theory, Springer (1998)

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

[10] J. Humphries, Linear Algebraic Groups, Springer (1998)

[11] F. Kirwan, Complex Algebraic Curves, London Mathematical Society, Student Texts 23, (1992)

[12] D. Marker. Zariski Geometries, Model Theory and Algebraic Geometry, Lecture Notes in Mathematics 1696, Springer (1998)

[13] J. Milne, Etale Cohomology, Princeton UP (1980)

[14] J. Milne, Lectures on Etale Cohomology, www.lsa.umich.edu/jmilne (1988)

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

[16] Z. Robinson. A Rigid Analytic Approximation Theorem, Asterisque 264 (2000)

Mathematics Department, The University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh, EH9 3JZ

  E-mail address: depiro@maths.ed.ac.uk