ZARISKI STRUCTURES AND ALGEBRAIC CURVES

TRISTRAM DE PIRO

Abstract. The purpose of this paper is to provide an alternative account of multiplicity for finite morphisms between smooth projective varieties. Traditionally, this has been defined using commutative algebra. In model theory, a different approach to multiplicity was developed by Zilber using the techniques of non-standard analysis. Here, we first reformulate Zilber’s method in the language of algebraic geometry, using specialisations, and, secondly, show that, in classical projective situations for smooth algebraic curves, the two notions essentially coincide.

We will work mainly in the language of Weil’s Foundations, namely using varieties instead of schemes. K will denote a big algebraically closed field. L ⊂ K will denote a small algebraically closed field. By an affine variety V, we mean a closed subset of K^{n} in the Zariski topology. If V is irreducible, we denote the ring of regular functions on V by K[V] and the function field by K(V). If k ⊂ K is perfect, we say that V is defined over k if I(V), the radical ideal of functions vanishing on V is generated by polynomials with coefficients in k. Any irreducible affine variety V has a minimal field of definition k_{V} with the property that any automorphism fixes V setwise iff it fixes k_{V} pointwise. This is a classical result due to Weil, but is in fact a special case of a more general construction due to model theorists of canonical bases, see [3]. By a variety, we will mean a set V, a covering of subsets V_{1}, … V_{m} and for each i a bijection f_{i} : V_{i} → U_{i} with U_{i} an affine variety and such that for each 1 ≤ i, j ≤ m, U_{ij} = f_{i}(V_{i} ∩ V_{j}) is an open subset of U_{i} and f_{ij} = f_{j}f_{i}^{−1} is an isomorphism between the affine varieties U_{ij} and U_{ji}. A variety V then inherits a natural Zariski topology by declaring U ⊂ V open if for each i, f_{i}(U ∩ V_{i}) is open in U_{i}. For k ⊂ K, we will say that V is defined over k, if the data (U_{i}, U_{ij}, f_{ij}) is defined over k in the sense of affine varieties. We let P^{n}(K) denote n-dimensional projective space over K, that is K^{n+1}/~, where ~ is the equivalence relation on K^{n+1} \{¯0} given by (x_{0}, …, x_{n})~(y_{0}, … y_{n}) iff λ(x_{0}, …, x_{n}) = (y_{0}, …, y_{n}) for some λ ∈ K. Writing elements of P^{n}(K) in homogenous coordinates, (x_{0} : x_{1} : …: x_{n}), we have natural bijections f_{i} between K^{n} and P^{n}(K)_{i} = {x̄ : x_{i} ≠ 0}. This gives P^{n}(K) the structure of a variety defined over the prime subfield and an induced Zariski toplogy. By a projective variety V, we mean a closed subset of P^{n}(K), using the coordinate charts f_{i}, V automatically is a variety in the sense defined above. Equivalently, a projective variety V is defined by a set of homogenous polynomials in K[x_{0}, … x_{n}] and is defined over k if the ideal I(V) is generated by homogenous polynomials with coefficients in k. If a variety is V defined over k and k ⊂ L ⊂ K with L algebraically closed then we will use the notation V (L) to denote V considered as a variety over L. In this case, we will require that a subvariety of V (L) is defined over L. We will use the notation X ×_{Y} Z to denote the fibre product of two varieties X and Z over Y. Given a variety V defined over k and a tuple of elements ā ∈ V^{n}, we will use k(ā) to denote the field of definition of ā. In the case when X = Spec(L), corresponding to an L rational point j : Spec(L) → Z;

The author was supported by the Seggie Brown Research Fellowship.

L ×_{Z} Y ⟶^{i} X
↓^{pr}  ↓^{j}
Y ⟶^{f} Z

we will often use the notation L ×_{Z} Y to denote the geometric fibre f^{−1}(y) of a point y ∈ Z, considered as a variety over L. Similar notation will be used in the case of sheaves. Given varieties Y,Z and a morphism g : Y → Z, we define the pullback of a coherent sheaf F on Z to be the sheafification of

g^{∗}F = O_{Y} ⊗_{g^{−1}O_{Z}} g^{−1}F

where g^{−1}F (U) = lim_{→,g(U)⊂V} F (V). Again, in the case when j : Spec(L) → Z is an L rational point and F is a coherent sheaf on Z, j^{−1}F = F_{z},the localised sheaf at z, and L ⊗_{O_{z,Z}} F_{z} is a vector space over L which, by slight abuse of notation, corresponds to the fibre of the sheaf F at z. Given a morphism f : X → Y, we let Ω_{X/Y} denote the sheaf of relative differentials on X. We will use the geometric construction of Ω_{X/Y} as Δ^{∗}J/J^{2} where Δ : X → X ×_{Y} X is the diagonal embedding and J/J^{2} is the normal bundle of Δ(X) in X ×_{Y} X. In the case when Y = Spec(L) for k ⊂ L ⊂ K and k the field of definition of X, we use the notation Ω_{X/L} to denote the sheaf of meromorphic differentials on X and Ω_{X/L}^{∗} the sheaf of meromorphic vector fields. There is a canonical isomorphism;

d : m_{z}/m_{z}^{2} → (Ω_{X/L})_{z} ⊗ L

d(f + m_{z}^{2}) = df

relating the sheaf of differentials to the cotangent space at a point. Using this isomorphism and Nakayama’s Lemma, one has that for an algebraic variety X of dimension n over k ⊂ L, Ω_{X/L} is a locally free module of rank n on the nonsingular locus U of X, see [6] for details.

1. Zariski Structures

Definition 1.1. Let (M, τ) be a topological space and let {C} denote the collection of closed sets on (M). We call (M, τ) a Zariski structure if the following axioms hold;

(L) Language: Basic relations are closed;

The diagonals Δ_{i} ⊂ M^{i} × M^{i} are closed.

Any singleton in M is closed.

Cartesian products of closed sets are closed

(P) Properness: The projection maps pr : M^{n+1} → M^{n} are proper and continuous, that is the images and inverse images of closed sets under pr are closed

(DCC) Descending Chain Condition: The topology given by the closed sets on M^{n} is Noetherian for all n ≥ 1. The condition (DCC) implies that every closed set C can be written uniquely (up to permutation) as a union of irreducible closed sets;

C = C_{1} ∪ … ∪ C_{n}

4 TRISTRAM DE PIRO

(DIM) Dimension: The following notion of dimension for closed sets C ⊂ M^{n} is well defined;

For irreducible C, dim(C) is the maximum m for which there exists a chain of irreducible closed sets C_{0} ⊂ C_{1} ⊂ … C_{m} = C.

For arbitrary closed C, dim(C) = max_{1≤i≤m}{dim(C_{i})} for C_{i} the irreducible components of C

(PS) Pre-Smoothness: For all closed irreducible sets C_{1}, C_{2} ⊂ M^{n}, with C_{1} ∩ C_{2} ≠ ∅,

dim_{comp}(C_{1} ∩ C_{2}) ≥ dim(C_{1}) + dim(C_{2}) − dim(M^{n}) (DF) Definability of fibres: If C ⊂ M^{n+m} is closed, then F (C, k) = {ā ∈ M^{n} : dim(C(ā)) ≥ k} is closed. (GF) Generic fibres: If C ⊂ M^{n+m} is closed and irreducible, then dim(C) = dim(pr(C)) + min_{ā∈pr(C)}dimC(ā)

Remarks 1.2. The definition of dimension easily implies the following properties;

(DU) Dimension of unions: For C_{1}, C_{2} closed, then dim(C_{1} ∪ C_{2}) = max{dim(C_{1}), dim(C_{2})} (DP) The dimension of a point is 0. (DI) Dimension of irreducible sets: If C_{1} ⊊ C_{2} and C_{2} is irreducible,

then dim(C_{1}) < dim(C_{2}).

We now show the following; Theorem 1.3. Let V be a smooth projective variety of dimension m defined over k and k ⊂ L with L algebraically closed, then V (L) considered as a toplogical space with closed sets given by the algebraic subvarieties defined over L is an irreducible Zariski structure of dimension m.

Proof. We will verify the axioms;

(L) We need only verify that the diagonals Δ_{i} ⊂ V^{i} × V^{i} are closed.

(P) An algebraic variety V is complete if for all varieties Y, the projection morphism

pr : V × Y → Y

is closed. Taking Y to be V^{n} in the above definition, complete varieties have the property that the projection maps

pr : V^{n+1} → V^{n}

are closed. If W ⊂ V is a closed subvariety of a complete variety V, then, as is easily checked, W is also complete. By assumption V is a closed subvariety of P^{N}(L) for some N. Now it is a classical fact that P^{N}(L) is complete, see for example [2].

(DCC) Let {W_{i} : i < ω} be an infinite descending chain of closed subvarieties of V^{n}. Let {U_{1}, … U_{n}} be an affine open cover of V^{n}. Then {U_{j} ∩ W_{i} : i < ω} defines a descending chain of closed subvarieties of each U_{j}. By the Nullstellensatz, each such chain stabilises inside U_{j}. Then clearly the chain stabilises inside V^{n}.

(DIM) For W an irreducible subvariety of V^{n}, we let dim_{geom}(W) = t.deg(L(W)/L). Then dim_{geom} corresponds to dim as defined above. To see this, suppose that dim(W) ≥ n + 1, and W is irreducible, then by definition one can find an irreducible closed subvariety W′ ⊂ W with dim(W′) ≥ n and so inductively dim(W′) ≥ n. Now take any affine open subset of V^{n} intersecting W′, so we may assume that W and W′ are affine as the function fields are unchanged. Let L[W] denote the coordinate domain of W, p the proper prime ideal corresponding to W′ and dim_{Krull} the Krull dimension of an integral domain. By Krull’s theorem, height(p) + dim_{Krull}(L[W]/p) = dim_{Krull}(L[W]), dim_{Krull}(L[W]) = t.deg(L(W)) and dim_{Krull}(L[W]/p) = t.deg(L(W′)), hence t.deg(L(W′))) < t.deg(L(W)). It follows that dim_{geom}(W′) < dim_{geom}(W) and so dim_{geom}(W) ≥ n+1. Conversely, if dim_{geom}(W) ≥ n + 1, then again assuming W is irreducible and affine, if we take f ∈ L[W] to be a non-unit, then each irreducible component of V (f) ⊂ W has codimension 1 in X, see [2]. Therefore, dim_{geom}(V (f)) ≥ n and inductively dim(V (f)) ≥ n. As each component of V (f) is a proper closed subset of X, dim(W) ≥ n+1. Now clearly we have that dim_{geom} corresponds to dim and so in particular we know that dim(V^{n}) = mn and the notion of dim on V^{n} is well defined.

(PS) A simple calculation shows that for (x_{1} … x_{n}) ∈ V^{n}, mx̄ ≅ Σ_{i=1}^{n}O_{x_{1}...x̂_{i}...x_{n}} ⊗ m_{x_{i}}. Hence,

T anx̄(V^{n}) = (mx̄/m2 ; x̄)^{∗} ≅ Σ_{i=1}^{n}(m_{x_{i}}/m_{x_{i}}^{2})^{∗} = Σ_{i=1}^{n}T an_{x_{i}}(V).

Therefore, V^{n} is smooth.

Now we use the following lemma;

Lemma 1.4. If X is a non-singular algebraic variety of dimension n, and Y, Z are irreducible closed subsets. Then if W is a component of Y ∩ Z, we have,

dim(W) ≥ dim(Y) + dim(Z) − n

or equivalently

codim(W) ≤ codim(Y) + codim(Z)

Proof. We have that Y ∩ Z ≅ Y × Z ∩ Δ(X) inside X ×_{L} X. Let g_{1}, …, g_{n} be uniformisers on an open subset U inside X. Then we saw above that Ω_{X/L} is just the pullback of the conormal sheaf J/J^{2} for the inclusion of Δ(X) inside X ×_{L} X. As Ω_{X/L} is locally free, so is J/J^{2}, and in particular generated freely on Δ(U) by the functions g_{1} ⊗ 1 − 1 ⊗ g_{1}, …, g_{n} ⊗ 1 − 1 ⊗ g_{n}. At a point x ∈ Δ(U), we have that g_{1} ⊗ 1 − 1 ⊗ g_{1}, …, g_{n} ⊗ 1 − 1 ⊗ g_{n} generate J_{x}/J_{x}^{2} and therefore form a basis for the vector space J_{x}/m_{x}J_{x} as clearly any function belonging to J_{x} lies in m_{x} the ideal of functions in O_{X×X,x} vanishing at x. Then, as J_{x}/m_{x}J_{x} is just the base change J ⊗ k(x) of the ideal sheaf J at the point x, it follows by Nakayama’s lemma that these functions generate J on an open neighborhood U containing x (not freely!). It follows that Y × Z ∩ Δ(X) is cut out by exactly n equations inside Y × Z, so by standard dimension theory we have the result.

□

It follows immediately that V^{n} satisfies (P S).

In order to check the final 2 axioms we introduce the following definitions;

Definition 1.5. If ā, b̄ ∈ V^{n} are tuples of elements, we define locus(ā/b̄) to be the intersection of all closed subvarieties defined over k(b̄) containing ā and locus_{irr}(ā/b̄) to be the intersection of all closed subvarieties defined over k(b̄)^{alg}. We define dim(ā/k) to be t.deg(k(ā)/k) and dim(ā/kb̄) to be t.deg(k(ā)/k(b̄)); if the underlying field k is clear from context, we will abbreviate this to dim(ā/b̄)

By the condition DCC, it is clear that locus and locus_{irr} are well defined. locus_{irr} is an irreducible subvariety of V^{n} containing ā, as if V is an irreducible components of locus_{irr} containing ā and k_{V} is the minimal field of definition, then k_{V} has only finitely many congugates under an automorphism fixing k(b̄)^{alg}, hence k_{V} ⊂ k(b̄)^{alg}.

Definition 1.6. If W ⊂ V^{n} is an irreducible closed subvariety, ā ∈ W, and b̄ is a tuple of elements such that k(b̄) contains a field of definition for W, then we say that ā is generic in W over b̄ if locus(ā/b̄) = W.

Lemma 1.7. Let W ⊂ V^{n} be an irreducible closed subvariety defined over k and ā generic in W over k. Then dim(W) = t.deg(k(ā)/k) = dim(ā/k).

Proof. By the above, dim(W) = dim_{geom}(W) = t.deg(k(W)/k). By choosing an open affine subvariety of W containing ā and defined over k, we can assume that W is affine. Now define a map ev : k[W] → k(ā) by setting ev(f) = f(ā). ev is injective as if f(ā) = 0, then as f has coefficients in k and ā is generic in W over k, f|W = 0. Clearly ev extends to a map on k(W) which is an isomorphism.

□

(DF) Let W ⊂ V^{n+m} be a closed subvariety and pr the projection onto n factors. We can cover (P^{N}(L))^{(n+m)} with finitely many affines of the form A^{N(n+m)}, hence we may assume that W is a closed subvariety of A^{N(n+m)} and show that Γ(ȳ) = {ā : dim(W (ā)) ≥ k + 1} is closed in pr(W). By additivity of t.deg and the lemma, this occurs iff we can find algebraically independent elements b_{1} … b_{k}b_{k+1} ⊂ b̄ ⊂ L such that W (b̄ā) holds iff

∃_{σ(k+2)} … ∃_{σ(Nm)}W (x_{1}, …, x_{Nm}, ā)

has maximal dimension for some permutation σ ∈ S_{Nm−(k+1)}. We may write each projection W_{σ} in the form

⋂_{i} F_{i}(x_{1} … x_{k+1}, ȳ) = 0 ∩ ⋂_{j} Q_{j}(x_{1} … x_{k+1}, ȳ) ≠ 0

where F_{i} and Q_{j} are polynomials in the variables x̄ȳ. Let θ_{σ}(ȳ) define the closed set given by the vanishing of all coefficients in the F_{i}. Then an easy calculation shows that Γ_{σ}(ȳ) = {ȳ ∈ pr(W) : θ_{σ}(ȳ)}, which is closed.

(GF) We first show the following;

Lemma 1.8. Let W ⊂ V^{n+m} be closed and irreducible, defined over k. Then āb̄ is generic in W over k iff ā is generic in pr(W) over k and b̄ is generic in W (ā) over k(ā).

Proof. One direction is straightforward, if ā is not generic in pr(W), then ā ∈ E ⊊ pr(W) and āb̄ ∈ pr^{−1}(E) ⊊ W. If b̄ is not generic in W (ā), then we can find X ⊊ W (ā) containing b̄ defined over k(ā). As we are working in a product of P^{m}(L), we can define X by a series of nhomogeneous equations with coefficients in k(ā). Applying Frobenius to these equations, we can in fact assume that the coefficients lie in k < ā >. Now a straightforward exercise in clearing denominators and writing affine equations in homogeneous form shows that we can write X as the fibre Y (ā) for some closed subvariety Y of V^{n+m}. Intersecting with W if necessary gives a proper closed Y ⊊ W with āb̄ ∈ W and defined over k.

For the other direction, suppose that āb̄ is not generic in W over k, then there exists X defined over k such that āb̄ ∈ X ⊊ W. Then ā ∈ pr(X) which is also closed and defined over k. Hence, pr(X) = pr(W). As b̄ ∈ X(ā), we have that dim(X(ā)) = dim(W (ā)) = m. By (DF),

X_{m} = {ā ∈ pr(X) : dim(X(ā)) = dim(W (ā)) = m}

is constructible and, by automorphism, can be seen to be defined over k. Hence, as ā was assumed to be generic, X_{m} is open inside pr(X). Now, using Lemma 1.7 and the hypotheses on ā,b̄, dim(X) ≥ dim(āb̄/k) = dim(b̄/āk) + dim(ā/k) = m + dim(pr(W)). However, choosing ā′b̄′ generic in W over k, we have that dim(W) = dim(ā′b̄′/k) = m + dim(pr(W)) by the properties of X_{k}. Hence, dim(X) ≥ dim(W) contradicting the fact that X ⊊ W and W was assumed to be irreducible.

□

Using the lemma, we can give an easy proof of (GF);

Let W ⊂ V^{n+m} be closed, irreducible and defined over k. Choose āb̄ generic in W over k. Then

dim(W) = dim(āb̄/k) = dim(b̄/āk) + dim(ā/k) = dim(pr(W)) + min_{ā∈pr(W)}W (ā).

The last equality follows from the previous lemma and (DF).

We have therefore checked all the axioms.

□

Definition 1.9. Given a closed subvariety W of V^{m}(L) and a closed F ⊂ W × V^{m}, all defined over k, we say that F is a cover of W if pr(F) = W and that ā ∈ W is regular for the cover if dimF (ā) = dimF (ā′) for ā′ generic in W over k.

2. Specialisations

For this section, we work in the language L_{rob} = (+, −, ×, 0, 1, Div) where, for an algebraically closed valued field (L, v, Γ) ,with value group Γ,we interpret the predicate Div by;

Div(x, y) ≡ v(x) ≤ v(y)

Abraham Robinson proves in [10] that T h(K) has quantifier elimination in L_{rob}.

In order to apply the technique of specialisations, we fix an algebraically closed field L with the trivial valuation v and construct a universal extension (K_{ω}, v_{ω}) as follows;

Set (K_{0}, v_{0}) = (L, v). Construct (K_{i+1}, v_{i+1}) inductively as follows;

Let K_{i}((t_{i+1})) be the field of formal Laurent series in the variable t_{i+1} over the algebraically closed field K_{i}. Define K_{i+1} = K_{i}((t_{i+1}))^{alg}. We can define a valuation v_{i+1} on K_{i}((t_{i+1})) extending the valuation v_{i} on K_{i} by setting

Given the tower of algebraically closed fields L ⊂ K_{1} ⊂ K_{2} ⊂ … ⊂ K_{i} ⊂ …, we set K_{ω} = ⋃_{i<ω} K_{i}

For all i < ω, K_{i+1} is equipped with a canonical valuation \overline{v_{i+1}} : K_{i+1} → Z defined as follows;

For f ∈ K_{i}((t_{i+1})), we set v_{i+1}(f) = ord_{i+1}(f), where ord_{i+1}(f) is the minimum n appearing in the Laurent expansion of f. As is shown in [7], (K_{i}((t_{i+1})), v_{i+1}) is the completion of (K_{i}(t_{i+1}), v), for the canonical valuation v on the function field K_{i}(t_{i+1}). It follows that K_{i}((t_{i+1})) is a Henselian field with respect to v_{i+1}. By Hensel’s lemma, K_{i}((t_{i+1}))^{alg} is a union ⋃_{i<ω} K_{i}((t_{i+1}^{1/n})) of ramified extensions of K_{i}((t_{i+1})). The valuation v_{i+1} extends uniquely to the spectral valuation on K_{i}((t_{i+1}^{1/n})) by the formula;

\overline{v_{i+1}}(α) = (1/n)v_{i+1}(N_{K_{i}((t^{1/n}))/K_{i}((t))}(α))

From the previous section, we have that P^{n}(L) and P^{n}(K_{ω}) with closed sets given by subvarieties defined over L and K_{ω} respectively are Zariski structures. We define a specialisation map π_{ω} : P^{n}(K_{ω}) → P^{n}(L) as follows. First, the maps;

π_{i+1,m} : P^{n}(K_{i}((t_{i+1}^{1/m}))) → P^{n}(K_{i})

are defined by

(f_{0} : …: f_{n}) ↦ (\overline{t^{s}f_{0}} : …: \overline{t^{s}f_{n}})

where s ∈ Z is chosen such that {t^{s}f_{0}, … t^{s}f_{n}} ⊂ O_{v_{i+1}} = K_{i}[[t_{i+1}^{1/m}]] and v_{i+1}(t_{i+1}^{s}f_{j}) = 0 for some j with 0 ≤ j ≤ n. Using the fact that the residue mapping is a homomorphism on K_{i}((t_{i+1}^{1/m})), this map is clearly well defined. Moreover, the maps π_{i+1,m} are compatible for m ∈ N, in the sense that, given m_{1} and m_{2}, for π_{i+1,m_{1}m_{2}} : P^{n}(K_{i}((t_{i+1}^{1/m_{1}m_{2}}))) → P^{n}(K_{i}), we have that π_{i+1,m_{1}m_{2}}|P^{n}(K_{i}(t_{i+1}^{1/m_{k}})) = π_{i+1,m_{k}} for k ∈ {1, 2}. Hence, the maps π_{i+1,m} naturally define a map π_{i+1} : P^{n}(K_{i+1}) → P^{n}(K_{i})

(f_{0} : …: f_{n}) ↦ π_{i+1,m}(f_{0} : …: f_{n})

where {f_{0}, …, f_{n}} ⊂ K_{i}[[t^{1/m}]]

Now, for M < N, let Π_{N,M} = π_{M+1} ∘ … ∘ π_{N} : P^{n}(K_{N}) → P^{n}(K_{M}) and let Π_{M,M} = Id. Then we have that for M_{1} ≤ M_{2} ≤ M_{3}, Π_{M_{1},M_{2}} ∘ Π_{M_{2},M_{3}} = Π_{M_{1},M_{3}}, hence the maps {Π_{N,M}} form an inverse system and we set Π_{ω} = ⋃_{M,N} Π_{M,N} : P^{n}(K_{ω}) → P^{n}(L).

We now show the following lemmas for the pair (P^{n}(K_{ω}), Π_{ω}).

Lemma 2.1. Let V ⊂ P^{n}(L) be a smooth projective variety defined over L. Then Π_{ω} : V (K_{ω}) → V (L) defines a homomorphism of Zariski structures, in the sense that for all closed W ⊂ V^{m} defined over L and ā ∈ W (K_{ω}), we have that Π_{ω}(ā) ∈ W (L).

Without loss of generality we can take V to be P^{n}(L) and consider the case m = 2. The Segre embedding is defined by;

P^{n}(L) × P^{n}(L) → P^{n(n+2)}(L)

((x_{0} : …: x_{n}), (y_{0} : …: y_{n})) ↦ (x_{0}y_{0} : …: x_{0}y_{n} : x_{1}y_{0} : …: x_{n}y_{n})

and the following diagram is easily checked to commute:

P^{n}(K_{i+1}) × P^{n}(K_{i+1}) ⟶^{Segre} P^{n(n+2)}(K_{i+1})
↓^{π_{i+1}×π_{i+1}}  ↓^{π_{i+1}}
P^{n}(K_{i}) × P^{n}(K_{i}) ⟶^{Segre} P^{n(n+2)}(K_{i})

Therefore, it is sufficient to prove that the property holds for π_{i+1} : P^{n(n+2)}(K_{i+1}) → P^{n(n+2)}(K_{i}). This is trivial to check using the fact that the residue map on K_{i}[[t^{1/m}]] is a ring homomorphism fixing K_{i}.

Now we show that (P^{n}(K_{ω}), Π_{ω}) has the following universal property;

Lemma 2.2. Let L ⊂ L_{m} be a field extension of transcendence degree m, V a smooth projective variety defined over L and suppose the map π : V (L_{m}) → V (L) is given satisfying the conclusion of Lemma 2.1. Then there exists an L-embedding α_{L} : L_{m} → K_{ω} with the property that Π_{ω} ∘ α_{L} = π.

Choose a transcendence basis {t_{1}, … t_{m}} for L_{m} over L. We may assume that V is P^{n}(L) for some L and that π : P^{n}(L_{m}) → P^{n}(L) is defined as above for some discrete valuation v on L_{m} with residue field L. Altering (t_{1}, … t_{m}) by automorphism if necessary, we may assume that v|L(t_{1}, …, t_{m}) is the canonical valuation given by;

v_{res} : L(t_{1}, …, t_{m}) → (Z^{m}, <)

v_{res}(t_{1}^{i_{1}} … t_{m}^{i_{m}}) = (i_{1} … i_{m})

where < denotes the lexographic ordering on Z^{m}. The completion of L(t_{1}, …, t_{m}) with respect to v_{res} is the formal Laurent series in m variables (L((t_{1}, …, t_{m})), \overline{v_{res}}). Let (L̂_{m}, v̄) be the completion of L_{m} with respect to v, then, as L((t_{1}, … t_{m})) is Henselian with respect to v̄_{res}, v̄ is the unique extension of v̄_{res} to L̂_{m}. Now, for 1 ≤ i ≤ m, there exist canonical isomorphisms between L((t_{i}, …, t_{m})) and L((t_{i}))((t_{i+1}, …, t_{m})). These combine to give an L-embedding α_{L} of L((t_{1}, …, t_{m})) into K_{m}. Moreover, an easy calculation shows that Π_{ω} ∘ α_{L} = π on L(t_{1}, …, t_{m}). Now, by the uniqueness of the valuation extension from α(L((t_{1}, …, t_{m}))) to K_{m} = α(L((t_{1}, …, t_{m})))^{alg}, for any extension of α to an embedding of L_{m} into K_{m}, we have Π_{ω}∘α_{L} = π on L_{m} as well.

3. Infinitesimal Neighborhoods

From now on, we fix a pair of Zariski structures and the specialisation map Π_{ω}, to give a triple ((V (L), V (K_{ω}), Π_{ω}) where V is a smooth projective variety defined over L.

Definition 3.1. For ā ∈ V (L)^{n}, we define the infinitesimal neighborhood of ā to be;

Vā = Π_{ω}^{−1}(ā)

The first property of infinitesimal neighborhoods is that we can move inside closed sets.

Lemma 3.2. If W (ȳ) is an irreducible closed set defined in V (L), b̄ ∈ W and dim(W) = r, then there exists a b̄′ ∈ Vb̄ ∩ W (K_{ω}) such that dim(b̄′/L) = r

Proof. Consider the collection of constructible sets inside V (L)^{n}

W (ȳ)∪{¬C(ȳ) : C closed, definable over L, dim(W (ȳ)∩C(ȳ) < r)}

As W is irreducible of dimension r, any finite subcollection has a realisation in V (L)^{n}. By compactness, we can find a realisation b̄′ in W (K) for L ⊂ K such that dim(b̄′/L) = r. It then follows that we can define a partial specialisation π : V (K) → V (L) by setting π(b̄′) = b̄, for if C(ȳ) is a closed set defined over L such that ¬C(b̄), then we must have that dim(W (ȳ) ∩ C(ȳ) < d otherwise, W being irreducible, W (ȳ) ⊂ C(ȳ), so by construction ¬C(b̄′) also holds. Now, using Lemma 2.2 applied to the field L(b̄′) which has transcendence degree r over L, we may assume that L(b̄′) ⊂ K_{r} ⊂ K_{ω} and the specialisation π is given by the restriction of Π_{ω}.

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We now come to the critical theorem, a more general version of which was originally proved by Zilber in the context of abstract Zariski structures;

Theorem 3.3. Suppose that F ⊂ D × V^{k} is an irreducible finite cover of D with D a smooth subvariety of V^{m}, and F, D defined over L, such that F (a, b). If a′ ∈ V_{a} ∩ D(K_{ω}) is generic in D over L, then we can find b′ ∈ V_{b} such that (a′, b′) ∈ F (K_{ω}).

We here only sketch the proof, full details may be found in [8]. We first consider the following collection of constructible sets defined over K_{ω}, with a′ ∈ V_{a} ∩ D(K_{ω}) generic over L;

{F (a′, y)} ∪ {¬C(d, y) : d ∈ V (K_{ω}), ¬C(Π_{ω}(d), b)}

As F is a finite cover and K_{ω} is algebraically closed, a realisation b′ of this collection lies in V (K_{ω}) and F (a′, b′) holds. Moreover, Π_{ω}(b′) = b, otherwise, as the diagonal x = y is closed, we have that b′ ≠ y is in the collection which is ridiculous.

If the collection is inconsistent, we find a closed set Q ⊂ V^{n+k} such that F (a′, y) ⊆ Q(d, y) whereas ¬Q(π(d), b).

The point of the smoothness assumption is to show that the parameter space

L(x, z) ⊂ D × V^{n} = {(x, z) : F (x, y) ⊂ Q(z, y)}

which in general is not relatively closed in D × M^{n} at least corresponds to a closed set over a dense open subset of D. More precisely, there is a closed subvariety P (x, z) ⊂ D × M^{n} and D′ ⊂ D, dim(D′) < dim(D), all defined over L, such that

1. P (x, z) ⊂ L(x, z).

2. L(x, z) ⊂ P (x, z) ∪ (D′ × V^{n}) (∗)

We have by assumption that L(a′, d) holds. As a′ was chosen to be generic over L and a′ ∉ D′, P (a′, d) holds. Applying the specialisation Π_{ω} gives that P (a, Π_{ω}(d)), hence F (a, y) ⊂ Q(Π_{ω}(d), y), hence Q(Π_{ω}(d), b) holds as well, contradicting the assumption.

Remarks 3.4. In fact the theorem can be improved to give the following more general result;

Suppose that F ⊂ D × V^{k} is an irreducible generically finite cover of D with D a subvariety of V^{m}. Then, if a ∈ D is a regular point for the cover and contained in the non-singular locus of D, a′ ∈ V_{a} ∩ D(K_{ω}) is generic in D over L, then we can find b′ ∈ V_{b} such that (a′, b′) ∈ F (K_{ω}).

4. Zariski Unramified Maps and Multiplicity

The purpose of introducing infinitesimal neighborhoods is to define an abstract notion of Zariski multiplicity.

Definition 4.1. Zariski multiplicity

Let hypotheses be as in Theorem 3.3

Given (a, b) ∈ F, set

mult_{ab}(F/D) = Card(F (a′, K_{ω}))∩V_{b}) for a′ ∈ V_{a}∩D generic over L

We want to show this is well defined.

Proof. Suppose a′′ ∈ V_{a} ∩ D with Card(F (a′′, K_{ω}) ∩ V_{b}) = n. Consider the relation N(x, y_{1}, …, y_{n}) ⊂ D × V^{nk}, given by

N(x, y_{1}, …, y_{n}) = F (x, y_{1}) ∧ … ∧ F (x, y_{n})

Then we have that N is a finite cover of D and, moreover, by smoothness of D, each irreducible component of N has dimension at least

n(dim(F) + (n − 1)k) − (n − 1)(dim(D) + nk) = dim(D) + n(n − 1)k − n(n − 1)k = dim(D)

so clearly each component is a finite cover of D. Now, choose an irreducible component N_{i} containing (a′′, b′′_{1}, …, b′′_{n}), so by specialisation also contains (a, b, …, b) and consider the open set U ⊂ N_{i} given by

U(x, y_{1}, …, y_{n}) = N_{i}(x, y_{1}, …, y_{n}) ∧ y_{1} ≠ y_{2} ≠ … ≠ y_{n}

Then, for a′ ∈ V_{a} generic in D, it follows we can find a tuple (b′_{1}, …, b′_{n}) such that N_{i}(a′, b′_{1}, …, b′_{n}), and (b′_{1}, …, b′_{n}) ∈ V_{(b,...,b)}. As is easily checked, the tuple (a′, b′_{1}, …, b′_{n}) is generic inside N_{i}, hence must lie inside U. This proves that the b′_{1}, …, b′_{n} are distinct, hence Card(F (a′, K_{ω}) ∩ V_{b}) ≥ n.

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Definition 4.2. We say that a point (ab) ∈ F is Zariski ramified if mult_{ab}(F/D) ≥ 2. Otherwise, we call such a point Zariski unramified.

Now suppose F ⊂ D × V^{n} is an irreducible finite cover of D with D smooth, then we have the following easily checked lemma

Lemma 4.3. mult_{a}(F/D) =_{def} Σ_{b∈F (a,L)}mult_{ab}(F/D) does not depend on the choice of a ∈ D, and is equal to the size of a generic fibre over D

A simple consequence is the following:

Lemma 4.4. If ā′ ∈ D(L), then F (ā′) contains a point of ramification in the sense of Zariski structures iff |F (ā′)| < |F (ā)| where ā is generic in D.

Proof. We have seen that |F (ā)| = Σ_{b̄∈F (ā′,L)}mult_{ā′,b̄}(F/D). If |F (ā′)| < |F (ā)|, then there must exist b̄ ∈ F (ā′) with mult_{(ā′,b̄)}(F/D) ≥ 2 so the result follows by the definition of ramification in Zariski structures. The converse is similar.

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We will also require the following results, that Zariski multiplicity is multiplicative over composition and preserved by open maps.

Lemma 4.5. Suppose that F_{1}, F_{2} and F_{3} are smooth, irreducible, with F_{2} ⊂ F_{1} × V^{k} and F_{3} ⊂ F_{2} × V^{l} finite covers. Let (abc) ∈ F_{3} ⊂ F_{1} × V^{k} × V^{l}. Then mult_{abc}(F_{3}/F_{1}) = mult_{ab}(F_{2}/F_{1})mult_{bc}(F_{3}/F_{2}).

Proof. To see this, let m = mult_{ab}(F_{2}/F_{1}) and n = mult_{bc}(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 V^{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 V^{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.

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Lemma 4.6. Let π_{1} : X → D and π_{2} : Y → D be covers with assumptions as in remarks following Theorem 3.3. Suppose moreover that there exist open smooth subvarieties U ⊂ X and V ⊂ Y and an isomorphism f : U → V such that π_{2} ∘ f = π_{1} on U. Then if a ∈ D is a regular point for the cover π_{1} and (ab) ∈ U, mult_{ab}(X/D) = mult_{af(b)}(Y/D).

Proof. We may assume that the open set U is maximal with the property that (ab) ∈ U and there exists an isomorphism with V ⊂ Y. Suppose mult_{ab}(X/D) = m. Then we can find a′ ∈ V_{a} ∩ D(K_{ω}) generic in D over L and b_{1}, …, b_{m} distinct such that X(a′b_{i}) holds for 1 ≤ i ≤ m. It will be sufficient to show that Y (a′f(b_{i})) holds and f(b_{i}) ∈ V_{f(b)}, for 1 ≤ i ≤ m, then, as f is injective, mult_{a,f(b)}(Y/D) ≥ m and the result follows by symmetry. By the fact that π_{2} ∘ f = π_{1} on U we clearly have that Y (a′f(b_{i})) holds. Let \overline{graph(f)} be the projective closure of the graph of f in the projective variety X × Y and π_{X}, π_{Y} the projections onto the coordinates X and Y. Then π_{X} satisfies the conditions of the remarks after Theorem 3.3, and moreover by assumption the point (ab) ∈ X is regular for the cover π and contained in the non-singular locus of X. Hence, we can find (cd) ∈ V_{af(b)} such that \overline{graph(f)}(a′b_{i}, cd) holds. As graph(f) is a 1−1 correspondence between U and V, if (a′b_{i}, cd) ∈ \overline{graph(f)} \ graph(f) then (a′b_{i}, cd) ∈ F_{X} ∪ F_{Y} where F_{X}, F_{Y} consist of the infinite fibres of the projections π_{X} and π_{Y} respectively. By (DF), both of these are defined over L and have dimension strictly less than graph(f). This contradicts the fact that (a′b_{i}, cd) is generic inside \overline{graph(f)} over L, hence (a′b_{i}, cd) ∈ graph(f) and as f is a bijection (cd) = (a′f(b_{i})). This shows that f(b_{i}) ∈ V_{f(b)} as required.

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5. Etale Morphisms and Algebraic Multiplicity

We review here the algebraic notions which will be required in the following section.

Definition 5.1. 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.

The notion of an etale morphism simplifies considerably when we assume that X and Y are smooth algebraic varieties over L, see [6];

Theorem 5.2. If X and Y are non-singular algebraic varieties over L and f : X → Y is a morphism, then f is etale iff df : (m_{x}/m_{x}^{2})^{∗} → (m_{f(x)}/m_{f(x)}^{2})^{∗} is an isomorphism everywhere.

Remarks 5.3. This gives us a convenient test for etaleness given an arbitrary morphism of finite type between smooth varieties X and Y. If we take local uniformisers g_{1}, … g_{n} at x ∈ X, the dg_{i} generate Ω_{X} freely on an open U′ of x. If we pull back a set of uniformisers f^{∗}f_{1}, …, f^{∗}f_{n} on Y to X, we can locally define the Jacobian Jacf̄ ; ḡ to be;

det(\frac{∂f^{∗}f_{i}}{∂g_{j}})

which means write the 1-forms f^{∗}df_{i} = Σ_{j}a_{ij}dg_{j} and take det(a_{ij}). If f is etale in a neighborhood of x, the f^{∗}df_{i} also generate Ω_{X} freely on an open U′′ of x. Taking the intersection U′′ = U ∩ U′, gives us that the Jacobian Jacf̄ ; ḡ|U′′ ≠ 0. Conversely, if Jacf̄ ; ḡ(x) ≠ 0, then it is non zero on an open neighborhood U′′ of x and by the above theorem we have that f is etale on this neighborhood.

We will also require some facts about the etale topology on an algebraic variety Y, see [5] 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 K, then all etale morphisms into Y must come from reduced schemes of finite type over K, 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 5.4. 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}^{∧}

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We define the Henselisation of a local ring R to be the smallest Henselian ring R′ ⊃ R, with R′ ⊂ F rac(R)^{alg}. We have in fact, see [5], that;

Theorem 5.5. Given an algebraic variety Y, O_{y,Y}^{∧} is the Henselisation of O_{y,Y}

Definition 5.6. Given smooth projective curves C_{1}, C_{2} and a finite morphism f : C_{1} → C_{2}, the algebraic multiplicity of f at a is ord_{a}(f^{∗}h) where h is a local uniformiser for C_{2} at f(a).

Remarks 5.7. This is independent of the choice of h, as the quotient of 2 uniformisers h/h′ is a unit in O_{f(a)}. Given finite morphisms f : C_{3} → C_{2} and g : C_{2} → C_{1}, if ord_{a,f(a)}(C_{3}/C_{2}) = m and ord_{f(a),gf(a)}(C_{2}/C_{1}) = n, then taking a local uniformiser h at gf(a), we have that g^{∗}h = h_{1}^{n}u locally at f(a) for a unit u and uniformiser h_{1} in O_{f(a)}. Similarily f^{∗}g^{∗}h = h_{2}^{mn}u′ for a unit u′ and uniformiserh_{2} in O_{a}. This shows that ord_{a,gf(a)}(C_{3}/C_{1}) = mn, so the branching number is also multiplicative for smooth projective curves.

Definition 5.8. Given smooth 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 5.9. 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, which generalises the result for curves;

Theorem 5.10. Algebraic multiplicity is multiplicative;

Given finite morphisms f : X_{3} → X_{2} and g : X_{2} → X_{1} between smooth projective varieties, for a ∈ X_{3} we have that

mult_{af(a)}(X_{3}/X_{2})mult_{f(a)gf(a)}(X_{2}/X_{1}) = mult_{agf(a)}(X_{3}/X_{1}).

Proof. The proof is an exercise in algebra, which we give for want of a convenient reference. First, the morphisms f and g are flat. This requires the following lemma, given as an exercise in [1], and the fact that smooth varieties are regular and Cohen-Macauley;

Lemma 5.11. Let f : X → Y be a morphism of varieties over L. Assume that Y is regular, X is Cohen-Macauley and that every fibre of f has dimension equal to dim(X) − dim(Y). Then f is flat.

Now we have a tower of local rings (R, m) ⊂ (S, n) ⊂ (T, o) with algebraically closed residue field L. Each extension is free by the flatness result and finiteness. For a finite free extension (R, m) ⊂ (S, n) of local rings, we also have the easily checked result that;

[S : R] = dim_{F r(R)}S⊗_{R}F r(R) = dim_{L}S⊗_{R/mR}R/mR = dim_{L}(S/mS)(∗).

For an extension (R, m) ⊂ (S, n) of local rings, we have that length(S/mS) = dim_{L}(S/mS), hence, by (*), the theorem reduces to checking that [T : R] = [T : S][S : R] which is standard, see [12].

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6. Equivalence of the Notions

This section is devoted to the main proofs of the paper. We show that, for morphisms between smooth projective varieties, the notions of etaleness and a Zariski unramified map essentially coincide (∗). We then show that, in the case of smooth algebraic curves (smooth projective varieties of dimension 1), algebraic multiplicity essentially coincides with Zariski multiplicity. Finally, we give a local version of (∗) for, possibly singular, algebraic curves. For the rest of the paper, we will often abbreviate smooth algebraic curves to just algebraic curves, as is done in [1], reserving the terminology singular algebraic curves to denote irreducible projective varieties of dimension 1.

Theorem 6.1. Let hypotheses be as in Theorem 3.3, with the additional assumptions that charL = 0 and F is smooth. Then F is a Zariski unramified cover of D iff F is an etale cover of D.

Let pr be the projection map of F onto D, then pr is a projective morphism. By Zariski’s Main Theorem, pr factors as a composition F →_{pr_{1}} F′ →_{pr_{2}} D with pr_{1} having connected fibres, pr_{1∗}O_{F} = O_{F′}and pr_{2} a finite morphism. The formal inverse pr_{1}^{−1} from F′ to F is a morphism corresponding to the identification of pr_{1∗}O_{F} and O_{F′}, hence pr_{1} is in fact an isomorphism. We may therefore assume that pr is a finite morphism.

Now suppose that pr is etale, then, pr is flat, see [6] for how this follows from Definition 5.1. As D is irreducible,

dim_{k(y)}(f_{∗}(O_{F}) ⊗_{O_{y}} k(y))

is independent of y ∈ D. As pr is etale, pr_{∗} : T_{x,F} → T_{pr(x),D} is an isomorphism, hence, by a simple calculation;

dim_{k(y)}(f_{∗}(O_{F}) ⊗_{O_{y}} k(y)) = |F (y)| for y ∈ D.

This shows that |F (y)| is independent of y ∈ D. By Lemma 4.4, this shows that pr is a Zariski unramified cover.

Conversely, suppose that pr is Zariski unramified. We first show that for generic ā ∈ D, |F (ā)| = deg(pr) = deg[k(F) : k(D)]. As char(k(F)) = 0, the extension is seperable so we can find a primitive element g ∈ k(F) such that k(F) = k(D)(g). Clearly the minimum polynomial p of g over k(D) has degree n = deg[k(F) : k(D)]. Let h_{1}, … h_{n−1} ∈ k(D) be the coefficients of p, then R(D)(h_{1} … h_{n−1}) determines the function ring of a Zariski open subset U of D. Clearly R(U)[g] is an integral extension of R(U) and corresponds to the projection restricted to U′ = pr^{−1}(U) ∩ g ≠ 0. By dimension theory, the zero set Z(g) ⊂ D cannot intersect with a generic fibre of the original map pr : F → D. Now we consider the discriminant D(p) of the polynomial p as a regular function on U and we have that for generic ā ∈ U that D(p)(ā) ≠ 0. This implies that for generic ā ∈ U |pr^{−1}(ā)| = n = deg[k(F) : k(D)]. Now we are in a position to apply Theorem 5, p145, of [11] which requires that D should be smooth, namely that pr_{∗} : T_{x,F} → T_{pr(x),D} is an isomorphism for x ∈ F. As both F and D were assumed to be nonsingular, this is sufficient to show that pr is etale by Theorem 5.2.

Remarks 6.2. When char(L) = p, the analogy fails. If we consider the Frobenius map F r : P^{1} → P^{1}, then Graph(F r) ⊂ P^{1} × P^{1} is a finite cover of P^{1}, which is smooth. The projection map pr onto the second coordinate is unramified in the sense of Zariski structures as pr is a bijection. However pr fails to be etale in the sense of algebraic geometry as pr_{∗} : T_{x,Graph(F r)} → T_{pr(x),P^{1}} is zero everywhere. However the following theorem shows that this is the only bad example and highlights one advantage of the Zariski method, namely that it is insensitive only to Frobenius. Theorem 6.3. Let hypotheses be as in Theorem 3.3, with the additional assumptions that char(L) = p ≠ 0 and F is smooth. Then, if F is an etale cover of D, F is a Zariski unramified cover. Conversely, if F is a Zariski unramified cover, then pr factors as a composition F →_{pr_{1}} F′ →_{pr_{2}} D in P roj with pr_{1} a purely inseperable connected cover and pr_{2} an etale cover. Proof. As in the previous theorem, we may assume that pr is a finite morphism. Suppose first that F → D is a finite morphism with F and D affine. We first find a field L such that k(F)/L is a purely inseperable extension and L/k(D) is seperable. Let R′ be the integral closure of R(D) in L and R′′ the integral closure of R(D) in k(F). As R(F) is integral over R(D) we have that R(F) ⊂ R′′, but F was assumed to be smooth so R(F) is integrally closed in k(F) and therefore R′′ = R(F). As the extensions k(D) ⊂ L ⊂ k(F) are finite algebraic, by [12], both R(F) and R′ are finite R′ and R(D) modules respectively. Therefore, corresponding to the ring inclusions

R(D) → R′ → R(F)

we have the sequence of finite morphisms

F →_{pr_{1}} Spec(R′) →_{pr_{2}} D

We first consider the cover F →_{pr_{1}} Spec(R′). Let g_{1}, … g_{m} generate R(F) over R′. As the extension k(F)/L is purely inseperable, we can write the minimum polynomials p_{i} of g_{i} in the form r_{i,0}g^{p^{n_{i}}} − r_{i,1} = 0 where r_{i,0} and r_{i,1} are in R′. As R(F)/R′ is finite, we can also find monic polynomials q_{i} with coefficients in R′ satisfied by g_{i}. Choose polynomials t_{i} = s_{i,0}x^{m_{i}} + s_{i,1}x_{m_{i}−1} + … s_{i,m_{i}} such that p_{i}t_{i} = q_{i}. By equating coefficients, we have that r_{i,0} = s_{i,0}^{−1} and r_{i,1}/r_{i,0} ∈ R′. Hence, we can take the p_{i} to be monic with coefficients in R′. As the p_{i} are minimal monic polynomials, we conclude that that R(F) is an extension of the form R′[g_{1}, …, g_{m}]/(g_{1}^{p^{n_{1}}} − λ_{1}, …, g_{m}^{p^{nm}} − λ_{m}) with λ_{i} ∈ R′. This is easily checked to be a connected cover of Spec(R′). In fact if we let θ = (F r^{−n_{1}}, …, F r^{−n_{m}}) ∘ (λ_{1} … λ_{m}), where the λ_{i} are considered as regular functions on Spec(R′) and F r^{−n_{i}} is the formal inverse Frobenius map, then the cover corresponds to the projection of Graph(θ) ⊂ Spec(R′) × A^{m} onto Spec(R′). As F was assumed to be smooth, Spec(R′) is a smooth seperable Zariski unramified cover of D. Applying the previous theorem, we conclude that Spec(R′) is an etale cover of D. Now, for the case when F and D are projective varieties, let U_{i} be an affine cover of D and R′(U_{i}) the corresponding normalisations. By uniqueness of integral closure, the R′(U_{i}) patch to form a cover F′ of D. In fact, by a classical result, see [6], we may assume that F′ is a smooth projective variety. As etaleness is a local condition for smooth varieties, the cover F′ is etale. Finally, check that the local maps pr_{1} : F_{i} → R′(U_{i}) patch on overlaps to give a morphism pr_{1} : F → F′. Clearly, this is an insperable connected cover, in fact if F′ is defined by the homogenous equations < f_{1}, … f_{n} > inside P^{N}, then F is isomorphic to the closed subvariety of P^{N} ×P^{m} defined by the extra equations < Y_{i}^{p^{n_{i}}}X_{N}^{j(i)} − λ_{i}(X_{0}, …, X_{N})Y_{0}^{p^{n_{i}}} > where 1 ≤ i ≤ m and j(i) is the degree of the polynomial λ_{i} in the affine coordinates P^{N}(L)_{i}.

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Remarks 6.4. We now show that the notions of Zariski multiplicity and algebraic multiplicity coincide for smooth algebraic curves when char(L) = 0, as usual with assumptions being as in Theorem 3.3., and find an analogous result when char(L) = p. Unfortunately, it does not seem possible to achieve this by counting points in the fibres, as in the previous theorems, so we need to find a local method. This will be the subject of the remainder of this section.

Remarks 6.5. We consider the case when F and D are smooth curves. Given a 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

24 TRISTRAM DE PIRO

unique (up to isomorphism) 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}}\}, 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 projective 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.

Theorem 6.6. Let hypotheses be as in Theorem 3.3, with the additional assumption that char(L) = 0 and F, D are smooth curves. Then the notions of Zariski multiplicity and algebraic multiplicity coincide.

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, 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, 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 [4];

Fact 6.7. 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 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) □

Remarks 6.8. Given the hypotheses of Theorem 6.6, 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 6.6 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. (†)

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 ∈ D is not wildly ramified for this cover, then we can follow through arguments (2) and (3) of Theorem 6.6. 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 6.6 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 in [9]. 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(C) admits a seperating transcendence basis over L, (see p27 of [1]). 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 [1]), 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 6.6 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. By a result in [6], (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, 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 [9].

Theorem 6.9. Let hypotheses be as in Theorem 6.6, 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. By Theorem 6.3, 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 [1]. By Remarks 6.8, the notions of Zariski multiplicity and algebraic multiplicity coincide for the morphism g. By Remarks 6.5, 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.

□

We now give local versions of Theorem 6.1 in the case of algebraic curves, where we allow F to be a singular algebraic curve, over a field L with char(L) = 0 and find an analogous version of Theorem 6.3, in the case when char(L) = p.

Theorem 6.10. Let hypotheses be as in Theorem 3.3, with the additional assumption that char(L) = 0 and D is a 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 6.6, the difference between the hypotheses there is that we do not assume that F is non-singular. Using the fact that D is smooth and the result of Theorem 6.6, we may, without loss of generality, assume that D = P^{1}(L). Now, one can follow through the proof of Theorem 6.6, 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, to show that a nonsingular open subvariety of an irreducible projective variety of dimension 1 is presmooth (see [13]). 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 non-singular open subvariety of F. Following the argument of Theorem 6.6, 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 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 6.11. Let hypotheses be as in Theorem 3.3, with the additional assumption that char(L) = p, D is a 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. The proof is similar to the previous theorem. By Remarks 6.8, we can assume that D = P^{1}(L). Using the fact that (ab) is nonsingular, one can either follow through the proof of Theorem 6.6, if (ab) is not wildly ramified for the cover, or one can use the method in

ZARISKI STRUCTURES AND ALGEBRAIC CURVES 31

[9], if (ab) is wildly ramified for the cover. For the second part of the theorem, one can use the same reasoning as in Theorem 6.10.

□

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