CHAPTER 0

PRELIMINARY MATERIAL

Paul Vojta

University of California, Berkeley

13 February 2021

This chapter gives some preliminary material on number theory and algebraic geometry.

Section 1 gives basic preliminary notation, both mathematical and logistical. Section 2 describes what algebraic geometry is assumed of the reader, and gives a few conventions that will be assumed here. Sections 3 and 4 give a few more details on schematic denseness, associated points, rational maps, rational sections of line sheaves, and how the above relate to the relation between Cartier divisors and line sheaves. Sections 5 and 6 discuss the field of definition of a closed subscheme and of a variety, respectively. Section 7 sets the basic notation and gives some fundamental results in number theory.

The remaining sections of this chapter describe some more specialized topics in algebraic geometry that will prove useful later: Kodaira’s lemma in Section 8, and descent in Section 9.

§1. General notation The symbols ℤ, ℚ, ℝ, and ℂ stand for the ring of rational integers and the fields of rational numbers, real numbers, and complex numbers, respectively. The symbol ℕ signifies the natural numbers, which in this book start at zero: ℕ = {0, 1, 2, 3, … }. When it is necessary to refer to the positive integers, we use subscripts: ℤ_{>0} = {1, 2, 3, … }. Similarly, ℝ_{≥0} stands for the set of nonnegative real numbers, etc.

The set of extended real numbers is the set \overline{ℝ} := \{−∞\} ∐ ℝ ∐\{∞\}. It carries the obvious ordering.

If k is a field, then k̄ denotes an algebraic closure of k. If α ∈ k̄, then Irr_{α,k}(X) is the (unique) monic irreducible polynomial f ∈ k[X] for which f(α) = 0 .

Unless otherwise specified, the wording almost all will mean all but finitely many. Numbers (such as Section 2, Theorem 2.3, or (2.3.5)) refer to the chapter in which

they occur, unless they are preceded by a number or letter and a colon in bold-face type (e.g., Section 3:2, Theorem A:2.5, or (7:2.3.5)), in which case they refer to the chapter or appendix indicated by the bold-face number or letter, respectively.

§2. Conventions and basic results in algebraic geometry It is assumed that the reader is familiar with the basics of algebraic geometry as given, e.g., in the first three chapters of [H 2], especially the first two. Note, however, that some conventions are different here.

This book will primarily use the language of schemes, rather than of varieties. The reader who prefers the more elementary approach of varieties, however, will often be able to mentally substitute the word variety for scheme without much loss, especially in the first few chapters.

With the exception of Appendix B, all schemes are assumed to be separated. Usually, schemes will also be assumed to be of finite type over some base scheme.

We often omit Spec when it is clear from the context; e.g., X(A) means X(Spec A) when A is a ring, ℙ_{A}^{n} means ℙ_{Spec A}^{n}, and X ×_{A} B means X ×_{Spec A} Spec B when A and B are rings.

The following definition gives slightly different names for some standard objects.

Definition 2.1. Let X be a scheme. Then a vector sheaf on X is a locally free sheaf of 𝒪_{X}-modules on X of constant finite rank. Its rank is the value of that constant rank (if X ≠ ∅). A morphism of vector sheaves is a morphism of 𝒪_{X}-modules on X. A line sheaf is a vector sheaf of rank 1 .

A vector sheaf ℱ of rank r on X can be described concretely by giving an open cover {U_{i}}_{i∈I} of X and isomorphisms ϕ_{i}: ℱ |_{U_{i}} →^{∼} 𝒪_{U_{i}}^{r} for all i ∈ I such that for all i, j ∈ I, the automorphism

ϕ_{i}|_{U_{i}∩U_{j}} ∘ ϕ_{j}^{−1}|_{U_{i}∩U_{j}}: 𝒪_{U_{i}∩U_{j}}^{r} → 𝒪_{U_{i}∩U_{j}}^{r}

is given by an element of GL_{r}(Γ(U_{i} ∩ U_{j}, 𝒪_{X})) . If 𝒢 is another vector sheaf on X, with corresponding open cover {V_{j}}_{j∈J} and isomorphisms ψ_{j}: 𝒢 |_{V_{j}} →^{∼} 𝒪_{V_{j}}^{s} , then a morphism of vector sheaves ℱ → 𝒢 is a morphism ρ: ℱ → 𝒢 of sheaves such that, for all i ∈ I and all j ∈ J , the morphism 𝒪_{U_{i}∩V_{j}}^{r} → 𝒪_{U_{i}∩V_{j}}^{s} corresponding to ρ|_{U_{i}∩V_{j}} via the isomorphisms ϕ_{i} and ψ_{j} is a linear homomorphism of 𝒪_{U_{i}∩V_{j}}-modules.

A line sheaf is also called an invertible sheaf by many authors.

Varieties Not all authors use the same definition of variety. Here we use the following definition.

Definition 2.2. Let k be a field. A variety over k, also called a k-variety, is an integral scheme, of finite type (and separated) over Spec k. If it is also proper over k, then we say it is complete. A curve is a variety of dimension 1 . A morphism of k-varieties is a morphism of schemes over Spec k. A subvariety (resp. open subvariety, closed subvariety) of a given variety over k is an integral subscheme (resp. open integral subscheme, closed integral subscheme) of that variety (with induced map to Spec k).

Note that, since k is not assumed to be algebraically closed, the set of closed points of a variety X over k is the set X(k̄) , modulo the action of Aut_{k}(k̄) . Also, the residue field k(P) for a closed point P will in general be a finite extension of k.

Remark 2.3. We have not assumed a variety to be geometrically integral. An advantage of this choice is that every irreducible closed subset of a variety will again be a variety, so that there is a natural one-to-one correspondence between the set of points of a variety and its set of subvarieties. This choice also agrees with the general philosophy that definitions should be weak. Since integrality is an absolute definition, if X is a variety over a number field k, then it is also a variety over any smaller number field k′, since one can compose X → Spec k with the map Spec k → Spec k′ to get X to be a scheme over k′. The latter is discussed more in Section 6.

In addition, we have (not just for regular field extensions):

Proposition 2.4. The association X ↦ K(X) induces an arrow-reversing equivalence of categories between the category of varieties and dominant rational maps over k, and the category of finitely-generated field extensions of k.

Proof. The proof of ([H 2], I Thm. 4.4) extends directly to the present case; see also ([EGA], I 7.1.16). □

Remark 2.5. Let k ⊆ L be fields. Then, for varieties (or schemes) X over k, the sets X(L) satisfy the following basic properties:

(a). 𝔸^{n}(L) = L^{n} in the obvious way. A similar statement holds for ℙ^{n}(L) . (b). A morphism f: X_{1} → X_{2} of schemes over k induces a natural map

f_{L}: X_{1}(L) → X_{2}(L) .

(c). If f is an immersion, then f_{L} is injective.

(d). If f is surjective and L is algebraically closed, then f_{L} is surjective.

(e). Since the field L has no nilpotents, X_{red}(L) = X(L) .

(f). An inclusion L_{1} ⊆ L_{2} of fields induces a natural injection X(L_{1}) ↪ X(L_{2}) . (g). If k′ is an extension field of k such that both k and L are contained in some larger field, then for schemes X over k there is a natural injection X(L) ↪ (X ×_{k} k′)(k′L) .

Note, however, that if L is not algebraically closed, then f_{L} in part (d) need not be surjective; consider for example f: 𝔸_{ℚ}^{1} → 𝔸_{ℚ}^{1} defined by z ↦ z^{2} .

We note that cohomology is geometric in nature: Let X be a scheme over k, let ℱ be a quasi-coherent sheaf on X, let L be a field containing k, let X_{L} = X ×_{k} L, let f: X_{L} → X be the projection morphism, and let ℱ_{L} = f^{∗}ℱ . Then, by the fact that cohomology commutes with flat base extension ([H 2], III Prop. 9.3),

H^{i}(X_{L}, ℱ_{L}) ≅ H^{i}(X, ℱ) ⊗_{k} L

for all i ≥ 0 .

Later chapters of the book will work extensively with schemes of finite type over the ring of integers of a number field. Such schemes have many of the same properties as varieties over a field. We describe one set of such properties here, generalizing a basic result on dimension in ([H], I Thm. 1.8A).

Proposition 2.6. Let f: X → Y be a morphism of finite type, where X and Y are noetherian integral schemes and Y is regular of dimension 1 . Assume that (i) Y has infinitely many points, or (ii) f is proper. Then: (a). For all closed points x of X, the point y := f(x) is a closed point of Y , and the induced extension k(x)/k(y) of residue fields is a finite extension.

(b). dim X = tr. deg K(X)/K(Y) + 1 ; and (c). X is catenary and equicodimensional; in particular,

(2.6.1)  dim \overline{\{x\}} + codim(x, X) = dim X

for all points x ∈ X. (Here equicodimensional means that all closed points of X have the same codimension in X.)

Proof. After passing to an open affine, we may assume that Y is affine, say Y = Spec A. Then A is a Dedekind domain.

We first prove (a).

Under assumption (i), A is not semilocal, hence is a Jacobson ring. The lemma then follows by the general Nullstellensatz ([E], Thm. 4.19).

Under assumption (ii), if x is a closed point, then so is y since by assumption f is proper, hence a closed morphism. Then x is a closed point of the fiber over y, which is a scheme of finite type over k(y) , and the assertion again follows from the general Nullstellensatz.

To prove (b), let x ∈ X and let y = f(x) . Let Spec A and Spec B be open affine neighborhoods of x and y, respectively, with f(Spec B) ⊆ Spec A, so that B is of finite type over A. Let 𝔭 and 𝔮 be the prime ideals of A and B corresponding to x and y, respectively. Since A is a regular ring, it is universally catenary by ([EGA], IV 5.6.4). Therefore B is also universally catenary, and by ([EGA], IV 5.6.1) we have

(2.6.2)  dim A_{𝔭} + tr. deg B_{𝔮}/A_{𝔭} = dim B_{𝔮} + tr. deg k(x)/k(y) .

Here the transcendence degree of an extension of entire rings means the transcendence degree of the corresponding extension of their fraction fields, so the second term equals tr. deg K(X)/K(Y) . If x is a closed point then dim A_{𝔭} = 1 and tr. deg k(x)/k(y) = 0 by part (a), so (2.6.2) becomes codim(x, X) = tr. deg K(X)/K(Y)+1 . Since this holds for all closed points x ∈ X, part (b) holds. It also implies that X is equicodimensional.

The above paragraph also shows that X is catenary. Finally, (2.6.1) follows from the fact that X is equicodimensional and catenary. Indeed, let

Z_{0} ⫋ Z_{1} ⫋ ⋯ ⫋ Z_{d} = X

be a maximal chain of irreducible subsets of X in which \overline{\{x\}} occurs. We then have d = codim(Z_{0}, X) = dim X since X is catenary and equicodimensional, respectively. (One can also obtain (2.6.1) from (2.6.2).) □

Extending ([E], Cor. 13.5), we then have:

Corollary 2.7. Let f_{1} : X_{1} → Y and f_{2} : X_{2} → Y be morphisms as above, and let g: X_{1} → X_{2} be a dominant morphism such that f_{1} = f_{2} ∘ g. Then the generic fiber of g has dimension dim X_{1} − dim X_{2} .

§3. Associated points, rational maps, and rational sections When dealing with schemes that are not necessarily reduced, we use a definition of rational map and rational section that is a bit different from the usual definition; see Definition 3.6. This definition is based on schematic denseness.

Definition 3.1. Let U be an open subset of a scheme X. We say that U is schematically dense in X if its schematic closure (the schematic image of the open immersion U ↪ X, or the smallest closed subscheme of X containing U ) is all of X. In this situation, we also say that U is an open dense subscheme of X.

Often it is convenient to think of schematic denseness in terms of the associated points of X.

Proposition 3.2. An open subset U of a noetherian scheme X is schematically dense if and only if it contains all associated points of X.

Proof. It suffices to show this when X is an affine scheme. Let A be the affine ring of X, let (0) = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{n} be a minimal primary decomposition of the ideal (0) in A, and let 𝔭_{i} = \sqrt{𝔮_{i}} for each i. Then 𝔭_{1}, …, 𝔭_{n} are the primes corresponding to the associated points of X.

First suppose that U contains all associated points of X. Let 𝔞 be the ideal corresponding to the schematic closure of U , and let f ∈ 𝔞. For all i, Spec A/𝔞 contains an open neighborhood of 𝔭_{i}; therefore (A/𝔞)_{𝔭_{i}/𝔞} = A_{𝔭_{i}} . In particular, f = 0 in A_{𝔭_{i}} , so Ann(f) meets A \ 𝔭_{i}. Thus, for all i there exists x_{i} ∈ Ann(f) such that x_{i} ∉ 𝔭_{i}. It follows by an easy exercise that there exists x ∈ Ann(f) such that x ∉ 𝔭_{1} ∪ ⋯ ∪ 𝔭_{n}. But then x is not a zero divisor, by ([L 2], X Prop. 2.9). This can happen only if f = 0 , so 𝔞 = 0 and thus U is schematically dense in X.

Conversely, suppose that U does not contain the point corresponding to 𝔭_{i}. After renumbering the indices, we may assume that i = n and that {i | 𝔭_{i} ⊉ 𝔭_{n}} = {1, …, r} for some r < n. Let

𝔞 = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{r}.

Since the chosen primary decomposition was minimal, we have 𝔞 ≠ (0) . We claim that U ⊆ Spec A/𝔞. Indeed, let 𝔭 be a prime ideal corresponding to a point in U . Since U is open and since 𝔭_{n} ∉ U , we have 𝔭 ⊉ 𝔭_{n}, and therefore 𝔭 ⊉ 𝔭_{i} for all i > r. Let S = A \ 𝔭. By ([A-M], Prop. 4.8(i)) and ([A-M], Prop. 4.9), we then have

(0) = ⋂_{i=1}^{n} S^{−1}𝔮_{i} = ⋂_{i=1}^{r} S^{−1}𝔮_{i} = S^{−1}𝔞.

Thus 𝔭 ∈ Spec A/𝔞 (since S^{−1}𝔞 = (0) implies 𝔞 ⊆ 𝔭), and (A/𝔞)_{𝔭} = A_{𝔭}. This holds for all 𝔭 ∈ U , so U is an open subscheme of Spec A/𝔞. This shows that U is not schematically dense in X. □

Proposition 3.3. Let U be an open dense subscheme of a noetherian scheme X. Then X is reduced, irreducible, or integral if and only if U has the same property.

Proof. The forward implication is trivial. If U is reduced, then X_{red} is a closed subscheme of X containing U , so X = X_{red} and therefore X is reduced. If U is irreducible, then so is X because the generic points of all irreducible components of X lie in U , by Proposition 3.2. □

Proposition 3.4. A locally noetherian, reduced scheme has no embedded points.

Proof. This is a well-known fact; see ([E], Exercise 11.10) for more details.

The question is local, so it is sufficient to consider the affine case, say X = Spec A. Since A is noetherian, its nilradical is the intersection of the minimal primes ([B], Ch. II § 2, Prop. 13). Since A is reduced, this gives a primary decomposition for the zero ideal.

□

As a consequence, for a locally noetherian scheme X, passing to X_{red} eliminates all the embedded points but leaves the underlying topological space unchanged.

Proposition 3.5. Let X and Y be schemes over a scheme S, with Y separated, and let f, g: X → Y be morphisms over S. If f and g agree on a schematically dense open subset of X, then f = g.

Proof. Let U be such an open subset. We have a morphism (f, g): X → Y ×_{S} Y ; the inverse image of the (closed) diagonal subscheme of Y ×_{S} Y is a closed subscheme of X containing U ; hence it is all of X and therefore (f, g) factors through the diagonal. This implies that f = g. □

Definition 3.6. Let X be a scheme.

(a). If X is noetherian, then a rational map f: X −−→ Y from X to another scheme Y is an equivalence class of pairs (U, f) , where U is a schematically dense open subset of X, f: U → Y is a morphism, and (U, f) is said to be equivalent to (U′, f′) if f and f′ agree on U ∩ U′.

(b). If X is noetherian, then a rational function on X is a class of pairs (U, f) , where U is a schematically dense open subset of X, f is a regular function on U , and the equivalence relation is similar to the one in part (a). (c). A rational section of a sheaf ℱ on X is a section of ℱ over a schematically

dense open set. (d). A rational section of a line sheaf ℒ on X is said to be invertible if it has

an inverse over a schematically dense open subset.

Remark. The relations in parts (a) and (b) above are equivalence relations because Y is separated, and because the open sets in question are schematically dense.

When working with rational maps, the following results are often useful.

Proposition 3.7. Let S be a locally noetherian scheme, let X be a normal scheme, separated and of finite type over S, let Y be a proper scheme over S, and let f: X −−→ Y be a rational map over S. Then the set of points in X where f is not defined has codimension ≥ 2 .

Proof. See ([EGA], II 7.3.6 and 7.3.7). □

Corollary 3.8. Let k be a field, let X be a normal curve over k, and let Y be a proper scheme over k. Then any rational map X −−→ Y over k extends to a morphism.

Proof. Immediate. □

The following proposition makes rigorous the intuition that, in most common cases, the points outside of a given open affine are “at infinity.”

Proposition 3.9. Let X be an integral scheme, separated and of finite type over an affine noetherian scheme Y = Spec B. Let U = Spec A be an open affine subset of X. Then, for any point P ∈ X with P ∉ U , there is some function f ∈ A which is not regular at P . Moreover, f can be chosen from among any given generating set of A over B.

Proof. Let {x_{1}, …, x_{n}} be a generating set for A over B, and suppose that all x_{i} are regular at P . Then there is an open affine neighborhood U′ = Spec A′ of P such that all x_{i} are regular on U′. Hence x_{1}, …, x_{n} all lie in A′, so A ⊆ A′. This gives a morphism ϕ: U′ → U which coincides with the identity map on U ∩ U′. Since X is separated, ϕ equals the identity on the closure of the dense set U ∩ U′, hence ϕ is an inclusion map U′ ↪ U . In particular U′ ⊆ U and hence P ∈ U , a contradiction. Thus if P ∉ U then some x_{i} is not regular at P . □

In general, the generic fiber of a morphism is not a subscheme. However, it is often dense in the scheme, in a way that is made precise below. We now provide counterparts to Propositions 3.2, 3.3, and 3.5, in this situation.

Definition 3.10. A morphism is scheme-theoretically dominant if its schematic image is equal to its codomain.

A morphism f: X → Y is scheme-theoretically dominant if and only if the sheaf ker(𝒪_{Y} → f_{∗}𝒪_{X}) contains no nontrivial quasi-coherent subsheaves. See [Stacks 01R6].

Proposition 3.11. Let Y be a regular integral noetherian scheme of dimension 1 , let k = K(Y) , let π: X → Y be a separated morphism of finite type, and let X_{k} denote the generic fiber of π. Then the following conditions are equivalent. (i). The map X_{k} → X is scheme-theoretically dominant; (ii). π is flat; (iii). X has no associated points outside of the generic fiber of π. Now assume that the above conditions hold. Then X is reduced, irreducible, or integral if and only if X_{k} has the same property.

Proof. We first show that (iii) ⟹ (ii) ⟹ (i) ⟹ (iii). The first implication follows from ([H 2], III Prop. 9.7).

Assume that X is flat over Y . Since flatness and schematic images are both local, we may assume that X is affine, say X = Spec A, and that X lies over an open affine Spec B of Y . The field of fractions of B is again k. Showing that X is the schematic image of X_{k} → X is equivalent to showing that A → A ⊗_{B} k is injective. But this follows by applying the flatness of A over B to the injection B ↪ k. Thus (i) holds.

Next assume that (i) holds, and that X has an associated point ξ outside of the generic fiber. Let Y′ = Y \ {π(ξ)}, and let X′ = X ×_{Y} Y′. Then X′ is an open subscheme of X. Let X′′ be the schematic closure of X′ in X. Then X′′ is a closed subscheme of X with the same generic fiber, so by (i), X′′ = X. But, by Proposition 3.2, ξ ∉ X′′, a contradiction. Thus (iii) holds.

Finally, assume that (i)–(iii) hold. If X_{k} is irreducible, then X has only one associated point by (iii), so it too is irreducible. The converse is clear. If X_{k} is reduced, then X_{k} → X factors through the closed subscheme X_{red} of X, so by (i) X is reduced. Again, the converse holds trivially. Therefore the last assertion holds. □

Proposition 3.5 can be generalized as follows.

Proposition 3.12 ([EGA], I 9.5.6). Let X and Y be schemes over a scheme S, with Y separated over S. Let f, g: X → Y be morphisms over S. If there exist a scheme W and a scheme-theoretically dominant morphism h: W → X such that f ∘ h = g ∘ h, then f = g.

Proof. Since Y → S is separated, the diagonal Δ is a closed subscheme of Y ×_{S} Y . Therefore its inverse image under the map (f, g): X → Y ×_{S} Y (equal to the product Y ×_{Y ×_{S}Y} X; ([EGA], I 4.4.1)) is a closed subscheme Z of X. Since f ∘ h = g ∘ h, the map (f, g) ∘ h = (f ∘ h, g ∘ h): W → Y ×_{S} Y factors through Δ . Therefore, by the universal property of the product, h factors through Z.

\xymatrix{W \ar[drr]^{h} \ar@{.>}[ddr]_{f∘h=g∘h} \ar@{.>}[dr] \\  & Z = Y ×_{Y ×_{S}Y} X \ar[d] \ar[r] & X \ar@{.>}[dl]_{θ} \ar[d]^{(f,g)} \\  & Y \ar[r]^{Δ} & Y ×_{S} Y}

This implies that Z contains the scheme-theoretic image of h, so Z = X. Thus, (f, g) factors through Δ ; i.e., there is a map θ: X → Y such that (f, g) = Δ ∘ θ. Let p_{1}, p_{2} : Y ×_{S} Y → Y be the projection maps. Then f = p_{1} ∘ (f, g) = p_{1} ∘ Δ ∘ θ = θ and similarly g = θ; hence f = g. □

As corollaries, we have the following.

Corollary 3.13. Let Y , π: X → Y , and k be as in Proposition 3.11, and assume that X satisfies conditions (i)–(iii) of the proposition. Let X′ be a separated scheme over Y . If f, g: X → X′ are morphisms over Y that agree on X_{k}, then f = g.

Proof. This is immediate from Proposition 3.12, since X_{k} → X is scheme-theoretically dominant. □

Corollary 3.14. Let X and Y be varieties over a field k, and let f, g: X → Y be morphisms over k. If f and g agree on a Zariski-dense set of points, then f = g.

Proof. Let Σ ⊆ X be a Zariski-dense set of points such that f(x) = g(x) for all x ∈ Σ . Let W be the scheme consisting of the disjoint union of Spec κ(x) for all x ∈ Σ , and let h: W → X be the morphism taking x ∈ Σ to x ∈ X, in the obvious way, for all x ∈ Σ . Since X is reduced and Σ is dense, h is scheme-theoretically dense. The assumptions imply that f ∘ h = g ∘ h, so f = g. □

§4. Cartier divisors and associated line sheaves

The above definition of invertible rational section of a line sheaf (Definition 3.6d) meshes well with the notion of a Cartier divisor.

Cartier divisors

We begin by recalling the definition of a Cartier divisor.

Definition 4.1. Let X be a scheme.

(a). The sheaf 𝒦 , called the sheaf of total quotient rings of 𝒪_{X} , is the sheaf associated to the presheaf U ↦ S^{−1}𝒪_{X}(U) , where S is the multiplicative system of nonzero elements which are not zero divisors. It contains 𝒪_{X} as a subsheaf of rings.

(b). Let 𝒪_{X}^{∗} denote the sheaf of invertible elements of 𝒪_{X} . Then a Cartier divisor on X is a global section of the sheaf 𝒦 ^{∗}/𝒪_{X}^{∗} .

The set of Cartier divisors forms a group, which is written additively instead of multiplicatively by analogy with Weil divisors.

By standard properties of sheaves, a Cartier divisor can be described by giving an open cover {U_{i}} of X and sections f_{i} ∈ 𝒦 (U_{i}) such that f_{i}/f_{j} lies in 𝒪_{X}(U_{i} ∩ U_{j})^{∗} for all i and j. Such a description is called a system of representatives for the Cartier divisor. If {(U_{i}, f_{i})}_{i∈I} is a system of representatives for a Cartier divisor D, then D|_{U_{i}} = (f_{i}) for all i. If the scheme X is integral, which is true in most applications, then 𝒦 is just the constant sheaf K(X) , and therefore in a system of representatives all f_{i} may be taken to lie in K(X) .

A Cartier divisor is principal if it lies in the image of the map

Γ(X, 𝒦 ^{∗}) → Γ(X, 𝒦 ^{∗}/𝒪_{X}^{∗}).

If X is integral, then this is equivalent to the assertion that all f_{i} can be taken equal to the same f ∈ K(X) . Two Cartier divisors are linearly equivalent if their difference is principal. A Cartier divisor is effective if it has a system of representatives {(U_{i}, f_{i})}_{i∈I} such that f_{i} lies in 𝒪_{X}(U_{i}) for all i. If D is a Cartier divisor, its support, denoted Supp D, is the set

Supp D = {P ∈ X | D_{P} ∉ 𝒪_{X,P}^{∗} }

(where D_{P} ∈ 𝒦_{P} denotes the germ of D at P ). Note that the group of units 𝒪_{X,P}^{∗} in the local ring 𝒪_{X,P} coincides with the stalk (𝒪_{X}^{∗})_{P} of the sheaf 𝒪_{X}^{∗} ⊆ 𝒦 . The support of D is a Zariski-closed subset of X. More concretely, if {(U_{i}, f_{i})}_{i∈I} is a system of representatives for D, then (Supp D) ∩ U_{i} equals the set of all P ∈ U_{i} such that f_{i} is not an invertible element of the local ring 𝒪_{X,P} at P . If D is a Cartier divisor with Supp D = ∅, then D = (1) .

For more information on Cartier divisors and how they compare to Weil divisors, see ([H 2], II §6).

Pull-backs of Cartier divisors A Cartier divisor cannot always be pulled back via a morphism; for example, if the support of the divisor contains the image of the morphism, then no such pull-back exists. A more precise criterion of when a pull-back exists can be found by looking at associated points.

Lemma 4.2. Let A be a commutative ring, let 𝔞 be an ideal of A, and let 𝔭_{1}, …, 𝔭_{r} be prime ideals of A with 𝔭_{i} ⊉ 𝔞 for all i. Then there exists f ∈ 𝔞 not lying in any of the 𝔭_{i}.

Proof. Exercise. See also ([B], Ch. II, § 1, Prop. 2). □

Lemma 4.3. Let A be a commutative noetherian ring, let S be the multiplicative system of nonzero elements of A which are not zero divisors, and let X = Spec A. (a). An element f ∈ A is a zero divisor or zero if and only if it is contained in some associated prime of A. Thus an element f ∈ A lies in S if and only if the principal open subset D(f) contains all associated points of X. (b). Let U be an open subset of X containing all associated points. Then U can be written as a union of principal open subsets D(f) with f ∈ S. (c). If U_{1} ⊆ U_{2} are open subsets as in part (b), then the map 𝒪_{X}(U_{2}) → 𝒪_{X}(U_{1})

is injective. (d). Let U be an open subset as in part (b). Then there is a natural injection

𝒪_{X}(U) → S^{−1}A. (e). Conversely, for any σ ∈ S^{−1}A, there is an open subset U of X such that σ

lies in 𝒪_{X}(U) (via the map of part (d)). (f). We have (S^{−1}A)^{∗} ⊆ A_{𝔭}^{∗} for all associated primes 𝔭 of A.

Proof. Part (a) is ([L 2], X Prop. 2.9).

Let 𝔞 be an ideal in A corresponding to the closed subset X \ U , let 𝔭_{0} be any prime in U , and let 𝔭_{1}, …, 𝔭_{r} be the associated primes of A. Then part (b) follows by applying Lemma 4.2 to 𝔞 and to 𝔭_{0}, …, 𝔭_{r}.

For part (c), assume the map is not injective and pick a nonzero section in 𝒪_{X}(U_{2}) restricting to zero in 𝒪_{X}(U_{1}) . By part (b), we may assume that U_{2} = D(f_{2}) for some f_{2} ∈ S, and that likewise U_{1} = D(f_{1}) . But, since f_{1} is not a zero divisor, the map A_{f_{2}} → A_{f_{1}} is injective, a contradiction.

Part (d) is easy to see, since we may find D(f) ⊆ U with f ∈ S, and then compose the injections 𝒪_{X}(U) → A_{f} → S^{−1}A.

For part (e), write σ = a/s with a ∈ A and s ∈ S; then we may take U = D(s) . Finally, for part (f), let s_{1}/s_{2} be an element of (S^{−1}A)^{∗}. Then s_{1}, s_{2} ∈ S, so by

part (a), s_{1}, s_{2} ∈ A \ 𝔭. Thus s_{1}/s_{2} ∈ A_{𝔭}^{∗}. □

Proposition 4.4. Let ϕ: X → Y be a morphism of noetherian schemes, and let D be a Cartier divisor on Y . If the support of D does not contain the image of any associated point of X, then D lifts to a Cartier divisor ϕ^{∗}D on X.

Proof. We may assume that X and Y are affine, say X = Spec B and Y = Spec A. Let S (resp. T ) be the multiplicative system of nonzero elements of A (resp. B) which are not zero divisors. We may assume that D is given on Y by an element f ∈ S^{−1}A. Let 𝔮_{1}, …, 𝔮_{r} be the associated primes of B, let 𝔭_{1}, …, 𝔭_{r} be their respective inverse images in A, and let 𝔭_{r+1}, …, 𝔭_{r′} be the associated primes of A. Let 𝔞 be the ideal in A given by

𝔞 = {a ∈ A : af ∈ A} .

Since the support of D does not contain any of the 𝔭_{i} (by assumption and by Lemma 4.3f), we have f ∈ A_{𝔭_{i}}^{∗} for all i, so 𝔭_{i} ⊉ 𝔞 for all i. By Lemma 4.2 there is an element s ∈ 𝔞 not lying in any of the 𝔭_{i}. Then sf ∈ A; also s ∈ S and ϕ^{∗}(s) ∈ T by Lemma 4.3a. Therefore f pulls back to the element ϕ^{∗}(sf)/ϕ^{∗}(s) ∈ T^{−1}B. Moreover, since f ∈ A_{𝔭_{i}}^{∗} for all i, we have sf ∈ A_{𝔭_{i}}^{∗} for all i, so in particular sf ∉ 𝔭_{i} for all i, and therefore ϕ^{∗}(sf) ∈ T . Thus f pulls back to an element of 𝒦 ^{∗}. □

The converse to the above proposition is false. For example, let ϕ: X → Y be the morphism of schemes corresponding to the ring homomorphism

k[x, y] → k[t, u]/(u^{2}, tu)

given by x ↦ t, y ↦ t, and let D be the Cartier divisor on Spec k[x, y] corresponding to x/y. Then the support of D contains the origin, which is the image of the embedded point in X, yet D lifts to the trivial divisor on X.

See also ([EGA], IV 21.4).

Cartier divisors and line sheaves

Definition 4.5. Let X be a noetherian scheme, let ℒ be a line sheaf on X, and let s be an invertible rational section of ℒ . Then, for any open affine subset U = Spec A of X over which ℒ is trivial, the restriction of s to U defines an element of S^{−1}A, where S is as in Lemma 4.3. Since s is invertible, this element actually lies in (S^{−1}A)^{∗}. This element depends on the choice of trivialization: changing the trivialization multiplies the element by an element of A^{∗}. Therefore, this gives a well-defined section of 𝒦 ^{∗}/𝒪^{∗} over U . These sections glue to give a global section of 𝒦 ^{∗}/𝒪^{∗} over X. This section, regarded as a Cartier divisor, is called the (Cartier) divisor of s, and is denoted (s) .

Proposition 4.6. Let X be a noetherian scheme.

(a). Let f be a rational function on X. Then f may be regarded as an invertible rational section of the trivial line sheaf 𝒪_{X} , and the definition of (f) as such coincides with the principal divisor (f) .

(b). Let ℒ be a line sheaf on X and let s be an invertible rational section of ℒ . Then s is a global (regular) section of ℒ if and only if (s) is effective. (c). Let ϕ: X′ → X be a morphism of noetherian schemes, let ℒ be a line sheaf on X, and let s be an invertible rational section of ℒ which is defined and nonzero at the images (under ϕ) of all associated points of X′. Then ϕ^{∗}s is an invertible rational section of ϕ^{∗}ℒ , and (ϕ^{∗}s) = ϕ^{∗}(s) .

(d). Let s_{1} and s_{2} be invertible rational sections of line sheaves ℒ_{1} and ℒ_{2} , respectively. Then s_{1} ⊗ s_{2} is an invertible rational section of ℒ_{1} ⊗ ℒ_{2} , and (s_{1} ⊗ s_{2}) = (s_{1}) + (s_{2}) .

Proof. These all follow immediately from the definition. □

Recall from ([H 2], II §6) the definition of the sheaf 𝒪(D) associated to a Cartier divisor D on a scheme X (Hartshorne denotes it ℒ (D) ). It is a subsheaf of 𝒦 .

Definition 4.7. Let D be a Cartier divisor on a noetherian scheme X. The global section 1 ∈ 𝒦 defines a rational section, called the canonical section of 𝒪(D) , and denoted 1_{D}.

In addition to ([H 2], II Prop. 6.13), we have the following properties.

Proposition 4.8. Let D be a Cartier divisor on a noetherian scheme X.

(a). If s is the canonical section of 𝒪(D) , then (s) = D.

(b). If ϕ: X′ → X is a morphism of noetherian schemes such that none of the associated points of X′ are taken into the support of D, then 𝒪(ϕ^{∗}D) ≅ ϕ^{∗}𝒪(D).

Proof. This is left as an exercise for the reader. □

Degrees of Cartier divisors on curves Let X be a nonsingular curve over a field k, and let D be a Cartier divisor on X. Since X is locally factorial, we may regard D as equivalent to a Weil divisor: D = ∑ n_{P} · P , where the sum is taken over all closed points P of X, n_{P} ∈ ℤ, and n_{P} = 0 for almost all P . Since k is not algebraically closed, the degree deg D is a little different from the definition in [H]:

Definition 4.9. With notation as above, the degree of D is the integer

deg_{k} D := ∑_{P} n_{P} · [k(P) : k] .

If the choice of k is clear from the context, then the subscript k may be omitted. For closed points P ∈ X, we also define the local degree:

deg_{k,P} D = n_{P} · [k(P) : k] ,

so that deg D = ∑ deg_{P} D.

Proposition 4.10. Let k be a field, let X and Y be nonsingular curves over k, let D be a divisor on Y , and let f: X → Y be a finite morphism. Recall that the degree of f is the degree [K(X) : K(Y)] of the corresponding extension of function fields. Then:

(a). For all closed points Q ∈ Y ,

(4.10.1)  ∑_{P ∈f^{−1}(Q)} deg_{P} f^{∗}D = (deg f)(deg_{Q} D) .

(b). deg f^{∗}D = (deg f)(deg D)

(c). If Y is a complete curve and f ∈ K(Y)^{∗}, then the degree of the principal divisor (f) is 0 .

Proof. By linearity, it suffices to prove part (a) when D is the single point Q with multiplicity 1 .

By ([EGA], II 7.4.11), X and Y may be embedded as open subschemes of nonsingular projective curves X′ and Y′, respectively, and (by Corollary 3.8) f extends to a morphism f′: X′ → Y′.

Let Spec A be an open affine neighborhood of Q in Y ; then f^{−1}(Spec A) is also affine, say f^{−1}(Spec A) = Spec B; moreover B is finite over A. We claim that all points in (f′)^{−1}(Q) lie in X. Indeed, suppose a point P ∈ (f′)^{−1}(Q) lies outside of X. Then it also lies outside of Spec B, so by Proposition 3.9 there exists a function f ∈ B which is not regular at P . Let

f^{n} + a_{n−1}f^{n−1} + ⋯ + a_{0} = 0

be an integral equation for f over A. Relative to the valuation corresponding to P , the first term of the above integral equation has (negative) valuation strictly smaller than the valuation of any of the other terms, leading to a contradiction. Thus the sum in (4.10.1) may be taken over all points of X′ lying over Q.

By ([EGA], II 7.4.18), the closed points of a complete nonsingular curve over k are in bijection with the valuation rings of the function field containing k; hence in (4.10.1) the sum may be taken over all valuations of K(X) extending the valuation of K(Y) corresponding to Q. In this form, the equation follows from ([B], Ch. VI, § 8, Thm. 2).

See also ([H 2], II Prop. 6.9).

Part (b) follows immediately from (a) by summing over all Q.

For part (c), if f ∈ k then (f) = 0 and the result is trivial. Otherwise, f determines a finite morphism f: Y → ℙ_{k}^{1} , and D is the pull-back of the principal divisor (z) on ℙ_{k}^{1} . Thus, part (c) follows by part (b) and the trivial computation deg(z) = 0 . □

§5. The field of definition of a closed subscheme In Weil-style algebraic geometry, a variety is an irreducible Zariski-closed subset of K^{n} or ℙ^{n}(K) for some algebraically closed field K, and we say that the variety is defined over a smaller field k if its defining ideal can be generated by polynomials with coefficients in k. This convention is also often used in the study of elliptic curves.

In the present language, this definition can be phrased (and generalized) as follows. See also ([EGA], IV 4.8).

Definition 5.1. Let k ⊆ k′ ⊆ k_{2} be fields, let X be a scheme of finite type over k, and let Z be a closed subscheme of X ×_{k} k_{2} . Then we say that Z is defined over k′ if there exists a subscheme Z′ of X ×_{k} k′ such that Z′ ×_{k′} k_{2} = Z. If so, then we also say that k′ is a field of definition for Z.

As was originally shown by Weil, in this situation there is a well-defined minimal field of definition.

Lemma 5.2. Let k ⊆ k_{2} be fields, let n ∈ ℕ, and let 𝔞 be an ideal in k_{2}[X_{1}, …, X_{n}] . Then there exists a field k_{1} such that k ⊆ k_{1} ⊆ k_{2} and, for all fields k′ with k ⊆ k′ ⊆ k_{2} , 𝔞 is generated by elements of k′[X_{1}, …, X_{n}] if and only if k′ ⊇ k_{1} . Moreover, k_{1} is finitely generated over k.

Proof. By ([C-L-O’S], Ch. 2, § 7, Prop. 6), every ideal in a polynomial ring over a field has a unique reduced Gröbner basis. Let k_{1} be the field generated over k by the coefficients of the elements of such a basis of 𝔞. Then the “if” part of the lemma is obvious, as well as the final assertion.

Conversely, suppose 𝔞 is generated by elements of k′[X_{1}, …, X_{n}] for some field k′. Let 𝔞′ be the ideal in k′[X_{1}, …, X_{n}] generated by those elements. Then 𝔞′ has a reduced Gröbner basis with coefficients in k′. But the definition of reduced Gröbner basis involves only linear algebra in the coefficients of the polynomials, so the unique reduced Gröbner basis is preserved by enlarging the field of coefficients. In particular the reduced Gröbner bases of 𝔞′ and 𝔞 coincide; hence k′ ⊇ k_{1} . □

Proposition 5.3. Let k ⊆ k_{2} be fields, let X be a scheme of finite type over k, and let Z be a closed subscheme of X ×_{k} k_{2} . Then there exists a unique minimal field of definition k_{1} of Z: for all fields k′ with k ⊆ k′ ⊆ k_{2} , Z is defined over k′ if and only if k′ ⊇ k_{1} . Moreover, k_{1} is finitely generated over k.

Proof. In the special case where X = 𝔸_{k}^{n}, this follows immediately by translating Lemma 5.2 into geometrical language. The general case follows by covering X by finitely many open affines U_{i}, which can then be regarded as closed subschemes of 𝔸_{k}^{n_{i}} . □

The following proposition gives a good idea of the structure of the field k_{1} .

Proposition 5.4. Let k be a field, let 𝔭 be a prime ideal in k̄[X_{1}, …, X_{n}] , and let k′ be the smallest field such that 𝔭 is generated by elements of k′[X_{1}, …, X_{n}] . Let L be the field of fractions of k̄[X_{1}, …, X_{n}]/𝔭, and let K be the subfield of L generated by k and the images of X_{1}, …, X_{n}. Then k′ is a purely inseparable extension of the algebraic closure of k in K; moreover, K is separable over k if and only if k′ is separable over k.

Proof. See ([W], Ch. 1, Prop. 23). □

Corollary 5.5. Let X be a variety over a field k, and let k′ be the minimal field of definition of X. Then k′ is a purely inseparable extension of the algebraic closure of k in K(X) ; moreover, K(X) is separable over k if and only if k′ is separable over k.

Proof. This follows by translating Proposition 5.4 into geometrical language. □

§6. The field of definition of a variety This section discusses what it means for a field to be a field of definition of a variety over a field, based on the concept of geometric integrality.

As noted in Remark 2.3, any variety over a number field k can also be given the structure of variety over ℚ, so saying that it is defined over k would run counter to intuition. For example, let X = Proj ℚ[x, y]/(y^{2} − 2x^{2}) . This is a closed subvariety of ℙ_{ℚ}^{1} , so it is a variety over ℚ, but intuition suggests that any “field of definition” of X should contain \sqrt{2} .

More abstractly, for any number field k, Spec k is a variety over ℚ (as well as over k). One expects that a “field of definition” of this variety would be k, and we show in this section that in fact it is, using the notion of geometric irreducibility. Note, however, that k need not be Galois over ℚ; therefore the field of definition of an abstract variety need not be Galois over the base field.

We begin by describing what happens to a variety under base change to a larger field.

Throughout this section, if K/k is a field extension, and if X is a scheme over k, then X_{K} denotes X ×_{k} K. Likewise, if A is an algebra over k then A_{K} denotes the K-algebra A ⊗_{k} K.

Lemma 6.1. Let k ⊆ K be fields.

(a). If A is an algebra over k, then the natural map A → A_{K} is injective. (b). If X is a scheme over k, then the natural projection X_{K} → X is surjective.

Proof. Part (a) follows from the fact that A is flat over k.

For part (b), we have that X_{K} is faithfully flat over X ([EGA], IV 2.2.3 and IV 2.2.13(i)), and therefore X_{K} → X is surjective ([EGA], IV 2.2.6). □

The following two subsections explore issues related to the following definitions.

Definition 6.2. A scheme X over a field k is geometrically irreducible (resp. geometrically reduced, or geometrically integral) if Xk̄ is irreducible (resp. reduced, or integral). All of these definitions are relative, in that they depend on k.

Separable field extensions and irreducible schemes A base change by a separable field extension may cause an irreducible scheme to become reducible (and, for an arbitrary field extension, any additional irreducible components occur already after base change to a separable subextension).

Proposition 6.3. Let X be a scheme of finite type over a field k, and let K be a purely inseparable field extension of k. Then the natural projection π: X_{K} → X induces a homeomorphism of the underlying topological spaces.

Proof. We may assume that X is affine, say X = Spec A. Let p = char k; we may assume p ≠ 0 .

We first claim that, for all f ∈ A_{K} , there exists e ∈ ℕ such that f^{p^{e}} ∈ A. Indeed, write f as a finite sum f = ∑ a_{i} ⊗ x_{i} with a_{i} ∈ A and x_{i} ∈ K for all i. The x_{i} generate a finite extension of k, and so for some e we have x_{i}^{p^{e}} ∈ k for all i. This value of e satisfies f^{p^{e}} ∈ A.

By Lemma 6.1b, X_{K} → X is surjective. Pick a prime 𝔭 of A and let 𝔮 be a prime of A_{K} lying over 𝔭. For f ∈ A_{K} , pick e ∈ ℕ as in the claim; then f ∈ 𝔮 if and only if f^{p^{e}} ∈ 𝔮, which holds if and only if f^{p^{e}} ∈ 𝔭. Thus

𝔮 = {f ∈ A_{K} : f^{p^{e}} ∈ 𝔭 for some e ∈ ℕ } .

In particular, X_{K} → X is injective.

It remains only to show that X_{K} → X is a homeomorphism; i.e., the topology of X_{K} is not strictly finer than that of X. To show this, it suffices to consider only principal open subsets of X_{K} . Let D(f) be such an open subset, and let e be as in the claim. Then D(f) = D(f^{p^{e}}) , so it comes from an open subset in X since f^{p^{e}} ∈ A. □

Proposition 6.4. Let X be a scheme of finite type over a field k. Then the following conditions are equivalent. (i). Xk̄ is irreducible. (ii). X_{k^{s}} is irreducible, where k^{s} denotes the separable closure of k. (iii). X_{K} is irreducible for all finite separable extensions K of k. (iv). X_{K} is irreducible for all field extensions K of k.

Proof. See ([H 2], II Ex. 3.15a) or ([EGA], IV 4.5.9). □

The following lemma gives some information on additional irreducible components that may arise from a base change.

Proposition 6.5. Let X be an irreducible scheme of finite type over a field k, and let K be an extension field of k. Then every irreducible component of X_{K} dominates X.

Proof. Since K is flat over k, the map X_{K} → X is also flat, so the going-down theorem ([E], Lemma 10.11) implies the lemma. □

Purely inseparable field extensions and reduced schemes Parallel to the situation in the previous subsection, a base change by an inseparable field extension may cause a reduced scheme to lose that property.

Proposition 6.6. Let X be a reduced scheme of finite type over a field k, and let K be a separable extension of k. Then X_{K} is also reduced.

Proof. We start with a lemma.

Lemma 6.6.1. Let A be an algebra over a field k, let K be a finite Galois extension of k, and let 𝔞 be an ideal in A_{K} . If 𝔞 is invariant under Gal(K/k) , then there is an ideal 𝔞_{0} of A such that 𝔞 = 𝔞_{0}A_{K} .

Proof. Write Gal(K/k) = {σ_{1}, …, σ_{n}}. By the Normal Basis Theorem, there exists x ∈ K such that σ_{1}(x), …, σ_{n}(x) is a basis for K over k. Let a ∈ 𝔞. We may write

a = ∑_{i=1}^{n} a_{i}σ_{i}(x) ,

with a_{i} ∈ A. It will suffice to show that the a_{i} actually lie in 𝔞.

We have

( σ_{1}(a) ; ⋮ ; σ_{n}(a) ) = ( \begin{matrix}σ_{1}(σ_{1}(x)) & … & σ_{1}(σ_{n}(x)) \\ ⋮ & ... & ⋮ \\ σ_{n}(σ_{1}(x)) & … & σ_{n}(σ_{n}(x)) \end{matrix} ) ( a_{1} ; ⋮ ; a_{n} ) .

By independence of characters, the matrix in the above equation is nonsingular. Hence the a_{i} can be written as linear combinations of the σ_{i}(a) with coefficients in K. This shows that the a_{i} all lie in 𝔞, as was to be shown. □

Returning to the proof of the proposition, the property of being reduced is local, so it suffices to consider affine schemes X = Spec A.

First consider the special case in which K is algebraic over k. Suppose that A_{K} contains a nonzero nilpotent element f . Then f lies in A_{K_{0}} for some finitely generated extension K_{0} of k, so we may assume that K is finite over k. After replacing K with its normal closure over k, we may assume that K is finite Galois over k. Then f ∈ A_{K} , K is a finite Galois extension of k, and of course f is still nonzero and nilpotent in A_{K} .

Since the nilradical of A_{K} is invariant under Gal(K/k) , Lemma 6.6.1 implies that it is generated by elements of A. In particular, X is not reduced, either, as was to be shown.

For the general case, it suffices to show that the proposition holds for a purely transcendental extension K/k, which is left as an exercise for the reader. □

Proposition 6.7. Let X be a scheme of finite type over a field k. Then the following conditions are equivalent. (i). Xk̄ is reduced.

(ii). X_{k^{perf}} is reduced, where k^{perf} denotes the perfect closure of k. (iii). X_{K} is reduced for all finite purely inseparable extensions K of k. (iv). X_{K} is reduced for all field extensions K of k.

Proof. See ([H 2], II Ex. 3.15b) or ([EGA], IV 4.6.1). □

Generalizing Proposition 6.5, we know that the associated points of a scheme do not go away after a base change to a larger field.

Proposition 6.8. Let X be a scheme of finite type over a field k, and let K be a field extension of k. Then the image of the set of associated points of X_{K} under the projection X_{K} → X is the set of associated points of X.

Proof. We may assume that X is affine, say X = Spec A. Recall that Ass_{A}(M) denotes the set of associated primes of an A-module M , and that Ass_{A}(A) equals the set of associated primes of A. By ([B], Ch. IV, § 2, No. 6, Exemple 4),

Ass_{A_{K}} (A_{K}) = ⋃_{𝔭∈Ass_{A}(A)} Ass_{A_{K}} (A_{K}/𝔭A_{K}) .

But by Lemma 6.1b the module A_{K}/𝔭A_{K} is nonzero for each 𝔭, so Ass_{A_{K}} (A_{K}/𝔭A_{K}) contains at least one prime of A_{K} over 𝔭. □

Fields of definition By making a base change to a larger field, a variety X may lose the property of being integral. This motivates the following definition.

Definition 6.9. Let X be a variety over a field k. We say that a field extension K of k is a field of definition for X if some irreducible component of X_{K} (with reduced induced subscheme structure) is geometrically integral.

Remark 6.10. In particular, k itself is a field of definition for X if and only if X is geometrically integral.

Lemma 6.11. Let X be a variety over k, let K be a field extension of k, and let U be a nonempty open affine subvariety of X. Then K is a field of definition for X if and only if it is a field of definition for U .

Proof. First assume that K is a field of definition for X. Then there exists an irreducible component X′ of X_{K} which is geometrically integral. Letting π: X_{K} → X denote the canonical projection, Proposition 6.5 implies that U′ := π^{−1}(U) ∩ X′ is a nonempty open subset of X′, which is therefore geometrically integral.

Conversely, if K is a field of definition for U , then π^{−1}(U) = U_{K} has an irreducible component U′ which is geometrically integral. Let X′ be the closure of U′ in X_{K} ; then X′ is an irreducible component of X_{K} .

To show that X′ is geometrically integral, we first show that X′ is geometrically irreducible. Let L be a field extension of K. Since X′ and U′ have the same generic point, the fibers of X′ ×_{K} L and U′ ×_{K} L over that point are the same, so (by Proposition 6.5) they have the same number of irreducible components. Therefore X′ ×_{K} L is irreducible for all L, so X′ is geometrically irreducible.

To show that X′ is geometrically reduced, let L be a purely inseparable field extension of K. By Proposition 6.3, X′_{L} is irreducible. Let π_{L}: X′_{L} → X′ be the projection map. Since U′ is geometrically reduced, U′_{L} = π_{L}^{−1}(U′) is reduced. By Propositions 6.8 and 3.2, U′_{L} is schematically dense in X′_{L}. Therefore X′_{L} is reduced by Proposition 3.3. □

Regular field extensions Definition 6.9 can be phrased in algebraic terms using the notion of regular field extensions.

Definition 6.12. A field extension K/k is regular if k is algebraically closed in K (i.e., any element of K algebraic over k is already contained in k), and K is separable over k.

Lemma 6.13. Let K/k be a field extension, and regard the algebraic closure k̄ of k as a subfield of \overline{K}. Then the following conditions are equivalent. (i). K is a regular extension of k; (ii). K is linearly disjoint from k̄ over k; and (iii). the natural map K ⊗_{k} k̄ → Kk̄ is injective.

Proof. The equivalence (i) ⟺ (ii) follows from ([L 2], Ch. VIII, Lemma 4.10). The equivalence of (ii) and (iii) is immediate from the definitions. □

Proposition 6.14. Let X be a variety over k. Then k is a field of definition for X if and only if K(X) is a regular extension of k.

Proof. By Lemma 6.11, we may assume that X is affine.

Let A be the affine ring of X, and let K = K(X) . Then K is the field of fractions of A, and Xk̄ is integral if and only if Ak̄ is entire.

First suppose that K is regular over k. Consider the composition of maps

A ⊗_{k} k̄ ↪ K ⊗_{k} k̄ → Kk̄ ,

where k̄ is viewed as a subfield of an algebraic closure \overline{K} of K. The first arrow is injective because k̄ is flat over k. The second arrow is injective by Lemma 6.13, so A ⊗_{k} k̄ is entire. Thus X is geometrically integral.

Conversely, assume that Ak̄ is entire. By Lemma 6.13, it will suffice to show that K is linearly disjoint from k̄ over k. Let θ_{1}, …, θ_{n} ∈ k̄, and assume that they are linearly independent over k. We need to show that they remain linearly independent over K.

Consider the composite map

(6.14.1)  A^{n} ⟶ Ak̄ ⟶ \overline{K} ,

where the first map takes (a_{1}, …, a_{n}) to ∑ a_{i} ⊗ θ_{i}. This first map is injective by flatness of A over k and by injectivity of the map k^{n} → k̄ determined by the linearly independent elements θ_{1}, …, θ_{n}. By incomparability ([E], Cor. 4.18), since Ak̄ is entire and integral over A, any map Ak̄ → \overline{K} extending the injection A ↪ K remains injective. Therefore the composite map (6.14.1) is injective, and thus θ_{1}, …, θ_{n} are linearly independent over K. □

Example 6.15 ([Mac], p. 384). This is an example of a field extension K/k which is not regular, even though k is algebraically closed in K.

Let F be a field of characteristic p > 0 , let k = F (x, y) with x and y indeterminates, and let K be the field of fractions of k[u, v]/(xu^{p} + yv^{p} − 1) .

We first claim that K is not separable over k. Indeed, u, v, 1 ∈ K are linearly dependent over k^{1/p} but not over k. Therefore K and k^{1/p} are not linearly disjoint over k, so by MacLane’s criterion K/k is not separable. (I thank Patrick Barrow for this shorter proof.^{1})

We next claim that k is algebraically closed in K. Suppose not, and let α be an element of K that is algebraic over k but does not lie in k. Since k(u) is purely transcendental over k, α is algebraic over k(u) , with the same degree and the same irreducible polynomial. Since K is purely inseparable over k(u) of degree p and since α ∈ K, α must also be purely inseparable over k(u) of degree p, and so α is purely inseparable over k of degree p. Thus K = k(α, u) , and therefore K = k(α)(u) is purely transcendental over k(α) . In particular it is separable, so since u, v, 1 ∈ K are linearly dependent over k(α)^{1/p}, they must also be linearly dependent over k(α) . The only linear dependence relation is x^{1/p}u + y^{1/p}v = 1 , so some scalar multiple of this relation must have coefficients in k(α) . Therefore x^{1/p} and y^{1/p} must lie in k(α) , a contradiction since [k(x^{1/p}, y^{1/p}) : k] = p^{2} > p = [k(α) : k] . Thus k is algebraically closed in K, as was claimed.

The following criterion is often useful for showing that a scheme is geometrically integral. I thank G. Rémond ([R], p. 7) and B. Poonen for useful conversations about it.

Proposition 6.16. Let X be an integral scheme of finite type over a field k. If X_{reg} contains a k-rational point, then X is geometrically integral.

Proof. Let A be the local ring at some regular, k-rational point on X, and let 𝔪 be its maximal ideal; then A/𝔪 = k. We first show that k is algebraically closed in the fraction field K(A) . Indeed, let α ∈ K(A) be an algebraic element over k. Then it is integral over A, hence it lies in A since the local ring is normal. Since α is congruent mod 𝔪 to an element of k, we may subtract that element and assume that α ∈ 𝔪. But now if α ≠ 0 then the same argument shows that α^{−1} lies in A, contradicting the fact that α cannot be a unit. Thus α ∈ k, so k is algebraically closed in K(A) .

To show that the field extension K(A)/k is separable, let d be the transcendence degree of K(A) over k. By ([E], Ch. 8, Thm. A), d = dim A, so by ([E], Ex. 16.15)

1Surprisingly, MacLane himself didn’t use his own criterion.

Ω_{A/k} is free of rank d over A. Then Ω_{K(A)/k} is a vector space of dimension d over K(A) , so by ([A], Cor. 16.17a), K(A) is separable over k. □

Remark 6.17. If k is perfect, then it suffices for X to have a normal rational point.

Example 6.18. Let k = ℚ, and let X = Spec k[x, y]/(y^{2}−2x^{2}) . Then X is integral but not geometrically integral, yet X(k) ≠ ∅. Therefore Proposition 6.16 is false without any conditions on the rational point.

More stuff

——— This is not written yet. However, I may need the following (which is useful in its own right).

Proposition 6.19. Let X be a variety over a field k, and let K be a normal field extension of k. Then all irreducible components of X_{K} are conjugate over k.

Proof. By Proposition 6.5, we may assume that X is affine, say X = Spec A. Write A = k[x_{1}, …, x_{n}]/P for some prime ideal P ; then X_{K} = Spec A_{K} , and

A_{K} ≅ K[x_{1}, …, x_{n}]/P K[x_{1}, …, x_{n}]

since K is flat over k. Note that the ring k[x_{1}, …, x_{n}] is integrally closed and that K[x_{1}, …, x_{n}] is the integral closure of k[x_{1}, …, x_{n}] in K(x_{1}, …, x_{n}) (since it is integrally closed and integral over k[x_{1}, …, x_{n}] ). Therefore, by ([Mat], Thm. 9.3(iii)), all primes of K[x_{1}, …, x_{n}] lying over P are conjugate over k(x_{1}, …, x_{n}) . By Proposition 6.5, these primes are exactly the primes corresponding to the irreducible components of X_{K} , so we conclude by noting that the restriction map

Gal(K(x_{1}, …, x_{n})/k(x_{1}, …, x_{n})) → Gal(K/k)

is bijective. □

§7. Conventions and basic results in number theory

It is assumed that the reader has mastered the basics of algebraic number theory as presented, for example, in Part I of [L 1], especially the first five chapters.

In addition, the following definitions are used.

Number fields

A number field k has a canonical set of places, denoted M_{k}. This set is in one-to-one correspondence with the disjoint union of:

(i). the set of real embeddings σ: k ↪ ℝ; (ii). the set of unordered complex conjugate pairs {σ, σ̄} of non-real embeddings

σ: k ↪ ℂ; and (iii). the set of nonzero prime ideals 𝔭 in the ring of integers R of k.

These places are referred to as real places, complex places, and non-archimedean places, respectively. In addition, an archimedean place is a real or complex place.

These places have norms ∥ · ∥_{v} defined by

∥x∥_{v} = \{ \begin{matrix}|σ(x)|, & if v is real, corresponding to σ: k ↪ ℝ; \\ |σ(x)|^{2}, & if v is complex, corresponding to σ, σ̄: k ↪ ℂ; \\ (R : 𝔭)^{− ord_{𝔭}(x)}, & if v is non-archimedean, corresponding to 𝔭 ⊆ R \end{matrix}

(In the last of the above three cases, ord_{𝔭}(x) denotes the order of x at 𝔭; i.e., the exponent of 𝔭 in the factorization of the fractional ideal (x) . This requires x ≠ 0 ; we also define ∥0∥_{v} = 0 .) These are not necessarily genuine absolute values, since the norm associated to a complex place does not satisfy the triangle inequality.

Instead of the triangle inequality, however, we have that if a_{1}, …, a_{n} ∈ k, then

(7.1)  ∥∑_{i=1}^{n} a_{i}∥_{v} ≤ n^{N_{v}} max_{1≤i≤n} ∥a_{i}∥_{v},

where

(7.2)  N_{v} = \{ \begin{matrix}1 & if v is real; \\ 2 & if v is complex; and \\ 0 & if v is non-archimedean. \end{matrix}

In particular, if v is non-archimedean, then ∥ · ∥_{v} obeys something stronger than the triangle inequality:

(7.3)  ∥x + y∥_{v} ≤ max(∥x∥_{v}, ∥y∥_{v}).

In addition, note that we have

(7.4)  ∑_{v∈M_{k}} N_{v} = [k : ℚ]

for any number field k.

The set of archimedean and non-archimedean places of a number field k are denoted M_{k}^{∞}(k) and M_{k}^{0} , respectively.

Function fields

Definition 7.5. Let F be a field and let n ∈ ℤ_{>0} . Then a function field over F in n variables is a finitely generated extension field of F of transcendence degree n.

Throughout this book, unless otherwise stated, function fields are assumed to be function fields in one variable. For function fields in one variable, the following strengthens Prop. 2.4:

Proposition 7.6. Let F be a field. Then the association X ↦ K(X) induces an arrowreversing equivalence of categories between the category of nonsingular projective curves with dominant morphisms over F , and the category of function fields over F in one variable with maps over F .

Proof. By ([EGA], II 7.4.18 and II 7.4.5), X ↦ K(X) maps onto the set of function fields over F (in one variable) (up to isomorphism). The sets of arrows in the two categories are equivalent by ([EGA], II 7.4.13).

Alternatively, the proof of ([H 2], I Cor. 6.12) extends readily to this case. □

Thus, a function field in one variable is the field k := K(X) of rational functions on some nonsingular projective curve X over a field F .

Fix a constant c > 1 . Then define a set M_{k} of places of k by defining, for each closed point P ∈ X, a place v with norm

(7.7)  ∥x∥_{v} := c^{−[k(P):F] ord_{P} x}

for all x ≠ 0 , where ord_{P} denotes the order of vanishing of the rational function x at P . (Again, we of course define ∥0∥_{v} = 0 for all v ∈ M_{k}.) It is customary to take c = e, so that log ∥x∥_{v} will take on integral values. Or, with finite fields, one could take c = #F , providing some parallels with the number field case.

In the number field case, there is really only one choice for M_{k}, and the normalizations of ∥ · ∥_{v} do not depend on additional choices. This is not the case in the function field case, however. Not only do the normalizations of ∥ · ∥_{v} depend on c and on F , but also the set M_{k} may vary for a given function field k. For example, with the field k = ℂ(X, Y) , we may take ℂ(X) as the field of constants and treat Y as an indeterminate, or vice versa. Therefore we will assume that, whenever a function field k is given, its set M_{k} of places is given, with norms at each of the places. Furthermore, an extension L/k of function fields is assumed to be one for which the set M_{L} is the set of all places extending places in M_{k}, and c and F coincide. Then, in the notation of the preceding paragraph, L = K(X′) for some curve X′ provided with a finite morphism to X.

For a function field k, all places v ∈ M_{k} are non-archimedean. Therefore, by (7.3), the set

{x ∈ k | ∥x∥_{v} ≤ 1 for all v ∈ M_{k} }

is a subfield of k, called the field of constants of k. It is a finite extension of F (indeed, if k′ is a maximal purely transcendental subextension of k/F , and k_{1} is the algebraic closure of F in k, then [k_{1} : F] ≤ [k : k′] ). Therefore we often specify just the set of places and the corresponding norms when giving a function field in one variable.

For a function field k, let N_{v} = 0 for all v ∈ M_{k}; then (7.1) and (7.2) hold also in the function field case. In addition:

Convention 7.8. If k is a function field, then we adopt the convention that [k : ℚ] = 0 .

With this convention, (7.4) holds for function fields as well.

As with number fields, for a function field k we let M_{k}^{∞} and M_{k}^{0} denote the sets of archimedean and non-archimedean places in M_{k}, respectively. Since all places in M_{k} are non-archimedean when k is a function field, this means that M_{k}^{∞} = ∅ and M_{k}^{0} = M_{k}.

Global and local fields Traditionally a global field is a number field or a function field in one variable over a finite field, and a local field is the completion of a global field at one of its places.

In this book, we are actually more interested in function fields of characteristic zero (mainly because Roth’s theorem is false in positive characteristic), so we rarely use the terms global field or local field. Instead of “global field,” we say number field or function field.

Likewise, instead of “local field,” we say completion of a number field or function field at one of its places. Although a given function field k may have places not in M_{k} (e.g., ℂ(X, Y) ), references to places of k will refer only to places in the chosen set M_{k}. If k is the completion of a number field or function field at a place v, then we let M_{k} = {v}.

The set of these (complete) fields is the disjoint union of:

(i). The set of finite extensions of ℚ_{p} for some rational prime p, (ii). The set of finite extensions of F ((T)) for some field F , and (iii). ℝ and ℂ. If k is a number field or function field and v ∈ M_{k}, then the norm ∥ · ∥_{v} extends

uniquely to a norm on the completion k_{v}. In that case, we may drop the subscript v, since the place is implicit from the fact that we are dealing with elements of k_{v}: ∥x∥ for x ∈ k_{v}.

If k is a number field, a function field, or the completion of such a field at one of its places, then we let \overline{M}_{k} denote the disjoint union of M_{L} for all finite extensions L of k in a fixed algebraic closure.

We have the following basic results from commutative algebra.

Proposition 7.10. Let A be a Dedekind ring, let k be its field of fractions, and let B be the integral closure of A in a finite extension L of k.

\begin{aligned}B & ⊆ & L \\ | & & | \\ A & ⊆ & k \end{aligned}

Then B is also a Dedekind ring. Also, if A is of finite type over a field, if char k = 0 , or if A is a complete local ring, then B is finite over A.

Proof. That B is Dedekind is a corollary of the Krull-Akizuki theorem ([B], Ch. VII, § 2, Cor. 2 of Prop. 5). Finiteness follows by ([B], Ch. IX, § 4, No. 1, Exemple), by ([B], Ch. IX, § 4, No. 1, Cor. of Prop. 1), and by ([B], Ch. IX, § 4, No. 2, Thm. 2). □

Note that the second assertion applies when k is a number field and A is its ring of integers, when k is a function field and A is the affine ring of some nonempty open affine in the associated projective curve, and when k is the completion of a number field or function field at one of its non-archimedean places and A is its valuation ring.

Note also that if L/k is inseparable, then all primes of A may ramify in B; an example is the extension 𝔽_{p}(T^{1/p})/𝔽_{p}(T) .

Proposition 7.11. Let k be a number field, a function field, or the completion of a number field or function field at one of its places. Let L be a finite extension of k, and let v ∈ M_{k}. Then

(7.11.1)  ∏_{\begin{matrix}w∈M_{L} \\ w|v \end{matrix}} ∥x∥_{w} = ∥N_{k}^{L}x∥_{v}  for all x ∈ L.

In particular,

(7.11.2)  ∏_{\begin{matrix}w∈M_{L} \\ w|v \end{matrix}} ∥x∥_{w} = ∥x∥_{v}^{[L:k]}  for all x ∈ k.

Proof. The first equation (7.11.1) is well known in number theory in the case when L is separable over k. In general it is due to Chevalley ([Ch], Ch. 4, Thm. 1). It is true by ([B], Ch. VI, § 8, No. 5, eqn. (9) and Cor. 2) and ([N], II 4.8):

∥N_{k}^{L}x∥_{v} = ∏_{\begin{matrix}w∈M_{L} \\ w|v \end{matrix}}∥N_{k_{v}}^{L_{w}}x∥_{v} = ∏_{\begin{matrix}w∈M_{L} \\ w|v \end{matrix}}∥N_{k_{v}}^{L_{w}}x∥_{w}^{1/[L_{w}:k_{v}]} = ∏_{\begin{matrix}w∈M_{L} \\ w|v \end{matrix}} ∥x∥_{w} .

The second equation follows easily from the first. □

Proposition 7.12 (Product Formula). Let k be a number field or function field. Then

∏_{v∈M_{k}} ∥x∥_{v} = 1 for all x ∈ k^{∗} .

Proof. First we note that, for a finite extension L/k of global fields, the Product Formula for k implies the Product Formula for L. Indeed, for all x ∈ L^{∗},

∏_{w∈M_{L}} ∥x∥_{w} = ∏_{v∈M_{k}} ∏_{\begin{matrix}w∈M_{L} \\ w|v \end{matrix}} ∥x∥_{w} = ∏_{v∈M_{k}}∥N_{k}^{L}x∥_{v} = 1

by (7.11.1). Thus it suffices to prove the Product Formula for ℚ and for F (T) . These are left to the reader as exercises. □

When k is a function field, Propositions 7.11 and 7.12 are equivalent to parts (a) and (c) of Proposition 4.10, respectively.

Artin and Whaples ([A], Ch. 12) showed that global fields (with a given collection of places and norms) are characterized by the product formula and certain “reasonableness” assumptions.

§8. Big divisors and Kodaira’s lemma

Some of the tools of classification theory in algebraic geometry are very useful also in diophantine geometry. One example of this is the definition of “big divisors” and “big line sheaves.”

We start with a lemma (which is useful to know in its own right).

Lemma 8.1. Let f: X′ → X be a finite morphism and let ℒ be an ample line sheaf on X. Then f^{∗}ℒ is also ample.

Proof. See ([H 1], Ch. I, Prop. 4.4). □

Theorem 8.2 ([I Thm. 10.2]). Let X be a complete variety over a field k, and let ℒ be a line sheaf on X. Assume that, for some integer m > 0 , ℒ^{⊗m} has a nonzero global section, and let m_{0} be the greatest common divisor of all such m. Then there is an integer d ∈ ℕ and constants c_{2} ≥ c_{1} > 0 such that

(8.2.1)  c_{1}n^{d} ≤ h^{0}(X, ℒ^{⊗nm_{0}}) ≤ c_{2}n^{d}

for all sufficiently large n > 0 .

Proof. This follows Iitaka’s proof quite closely, except that this proof allows X to be singular. By replacing ℒ with ℒ^{⊗m_{0}} , we may assume that m_{0} = 1 .

Let A be the set {m ∈ ℤ_{>0} : h^{0}(X, ℒ^{⊗m}) > 0}. Then A contains all sufficiently large integers. For each m ∈ A let Φ_{m}: X −−→ ℙ_{k}^{r_{m}} be a rational map associated to some basis of H^{0}(X, ℒ^{⊗m}) over k, and let W_{m} be the closure of its image. The function fields K(W_{m}) are subfields of K(X) containing k, and if m, m′ ∈ A then K(W_{m+m′}) ⊇ K(W_{m}) . Since K(X) is a finitely generated field extension of k, it is a straightforward exercise to show that any ascending chain of subfields of K(X) containing k must eventually stabilize; applying this to the set A ordered by m_{1} ≥ m_{2} if m_{1}−m_{2} ∈ A∪{0} (which is a directed system), it follows that K(W_{m}) is independent of m for all sufficiently large integers m. Fix one such integer m_{1} . Let d be the transcendence degree of K(W_{m_{1}}) over k.

We first show a lower bound for h^{0}(X, ℒ^{⊗nm_{1}}) , n ∈ ℤ_{>0} . Let {s_{0}, …, s_{r}} be the basis for H^{0}(X, ℒ^{⊗m_{1}}) over k used to define Φ_{m_{1}} ; then s_{i}/s_{j} is the element of K(X) corresponding to the ratio x_{i}/x_{j} of coordinate functions on W_{m_{1}} , for all i, j ∈ {0, …, r}. Since the map K(W_{m_{1}}) → K(X) is injective, the maps H^{0}(W_{m_{1}}, 𝒪(n)) → H^{0}(X, ℒ^{⊗nm_{1}}) are injective for all n. Since dim W_{m_{1}} = d, we then have

h^{0}(X, ℒ^{⊗nm_{1}}) ≥ h^{0}(W_{m_{1}}, 𝒪(n)) ≥ c_{1}n^{d}

for all n ≫ 0 , for some fixed c_{1} > 0 .

We next show a similar upper bound. Let X_{1} be the closure of the graph of Φ_{m_{1}} , let X_{2} → X_{1} be a morphism as in Chow’s lemma, and let X′ be the normalization of X_{2} . By Lemma 8.1, X′ is a projective variety. We therefore have a birational projective morphism α: X′ → X, and a morphism ϕ_{0} : X′ → W_{m_{1}} such that the following diagram commutes.

\xymatrix{X′ \ar[d]^{α} \ar[dr]^{ϕ_{0}} \ar[r]^{ϕ} & W′ \ar[d]^{β} \\ X \ar@{.>}[r]^{Φ_{m_{1}}} & W_{m_{1}}}

Since X′ is a projective variety, ϕ_{0} is a projective morphism, so we can apply Stein factorization to get ϕ_{0} = β ∘ϕ, with β finite and ϕ a morphism with connected fibers; moreover, ϕ_{∗}𝒪_{X′} = 𝒪_{W′} ([H 2], III Cor. 11.5 and proof). This latter fact also implies that the map Γ(W′, ℳ) → Γ(X′, ϕ^{∗}ℳ) is an isomorphism for all line sheaves ℳ on W′.

The remainder of this part of the proof will use the notation of divisors, often referring to Weil divisors. For Weil divisors D on X′, let D_{hor} (resp. D_{ver} ) denote the part of D composed of components that dominate W′ via ϕ (resp. components that are contained in fibers of ϕ); then D = D_{hor} + D_{ver} . Now fix a section s_{0} ∈ Γ(X, ℒ^{⊗m_{1}}) , and let D be the Cartier divisor (s_{0}) . We claim that, if n ∈ ℤ_{>0} and s ∈ Γ(X, ℒ^{⊗nm_{1}}) is a global section, then (α^{∗}s)_{hor} = n(α^{∗}D)_{hor} . Indeed, taking horizontal part commutes with addition (and subtraction) of divisors, and the difference (α^{∗}s) − α^{∗}D is the principal divisor generated by s/s_{0}^{n} ∈ K(X) = K(X′) . This rational function lies in K(W_{m_{1}}) , hence in K(W′) , so the above principal divisor is the pull-back of a divisor on W′. Therefore it has no horizontal part. Thus the fixed part of the pull-back of the complete linear system |nD| to X′ contains n(α^{∗}D)_{hor} , and therefore

(8.2.2)  {f ∈ K(X) : (f) + nD ≥_{c} 0} ⊆ {f ∈ K(X′) : (f) + n(α^{∗}D)_{ver} ≥_{w} 0} ,

where the inequalities ≥_{c} and ≥_{w} refer to the cones of effective Cartier divisors and effective Weil divisors, respectively. (The inclusion may be strict due to effective Weil divisors on X′ that do not come from effective Cartier divisors on X.)

Now, by Lemma 8.1, the pull-back of a hyperplane section of W_{m_{1}} to W′ is ample; therefore by ampleness there is a Cartier divisor E on W′ such that ϕ^{∗}E ≥_{w} (α^{∗}D)_{ver} . Hence

(8.2.3) \begin{aligned}\{f ∈ K(X′) : (f) + n(α^{∗}D)_{ver} ≥_{w} 0\} & ⊆ \{f ∈ K(X′) : (f) + nϕ^{∗}E ≥_{c} 0\} \\ & = \{f ∈ K(W′) : (f) + nE ≥_{c} 0\} ; \end{aligned}

note that since X′ is normal, an effective Weil divisor that is also Cartier is effective as a Cartier divisor ([H 2], II Prop. 6.3A). Also note that the second step holds because ϕ_{∗}𝒪_{X′} = 𝒪_{W′} . Combining (8.2.2) and (8.2.3), taking dimensions, and rewriting in terms of global sections of line sheaves gives

h^{0}(X, ℒ^{⊗nm_{1}}) ≤ h^{0}(W′, 𝒪(nE)) .

But the right-hand side is bounded by a constant multiple of n^{dim W′} = n^{d}. Indeed, we may add an effective divisor to E to make it very ample; then the bound follows by the theory of the Hilbert polynomial, or by Riemann-Roch.

Thus, (8.2.1) holds provided m_{1} | n (we are still assuming m_{0} = 1 ). To prove the general case, first note that there is a fixed integer a > 0 such that for all n, there is some q ∈ A such that q ≡ n (mod m_{1}) and such that am_{1} − q also lies in A. Then, for n ≫ 0 ,

c_{1}(n − q)^{d} ≤ h^{0}(X, ℒ^{⊗(n−q)})
≤ h^{0}(X, ℒ^{⊗n})
≤ h^{0}(X, ℒ^{⊗(n−q+am_{1})}) ≤ c_{2}(n − q + am_{1})^{d} ,

from which the general case of (8.2.1) follows readily. □

It is possible for m_{0} > 1 to occur; for example, consider a line sheaf ℒ corresponding to a nontrivial torsion point in Pic^{0}(X) .

The above integer d always satisfies d ≤ dim X. Indeed, pull back to a projective variety by Chow’s lemma, and write ℒ as 𝒪(D_{1} − D_{2}) with D_{1} ample and D_{2} effective. Then

h^{0}(X, ℒ^{⊗n}) ≤ h^{0}(X, 𝒪(nD_{1})) ≪ n^{dim X} .

In fact, the above bound holds for any projective scheme.

Definition 8.3. Let X, k, and ℒ be as above. Then the dimension (or ℒ -dimension) κ(ℒ) of ℒ is defined to be the integer d ≥ 0 in Theorem 8.2 if some positive tensor power of ℒ has a nonzero global section, and −∞ otherwise. If D is a Cartier divisor on X, then the dimension (or D-dimension) κ(D) of D is defined to be κ(𝒪(D)) .

Thus, the Kodaira dimension of a nonsingular complete variety is just the dimension of its canonical line sheaf.

Definition 8.4. Let X and k be as above. Then a line sheaf ℒ on X is big if κ(L) = dim X. Likewise, a Cartier divisor D on X is big if 𝒪(D) is big.

Big divisors often behave like ample divisors outside of a proper Zariski-closed subset. For example, Lemma 8.1 has the following counterpart for big line sheaves and generically finite morphisms.

Proposition 8.5. Let f: X′ → X be a generically finite morphism and let ℒ be a big line sheaf on X. Then f^{∗}ℒ is also big.

Proof. Indeed, for all n the map

f^{∗}: H^{0}(X, ℒ^{⊗n}) → H^{0}(X′, f^{∗}ℒ^{⊗n})

is injective. □

On projective varieties, big divisors can be easily characterized in terms of ample and effective divisors.

Proposition 8.6 (Kodaira’s lemma). Let X be a projective variety over a field k, and let D be a Cartier divisor on X. (a). If D is big, then for all divisors A on X, there is an integer n > 0 such

that nD − A is linearly equivalent to an effective divisor. (b). The divisor D is big if and only if some positive multiple of it is linearly

equivalent to a sum of an ample divisor and an effective divisor.

Proof. First suppose that D is big. The assertion depends only on the linear equivalence class of A, so we may assume that A = A_{1} − A_{2} , where A_{1} and A_{2} are effective Cartier divisors. Then it will suffice to show that nD − A_{1} is linearly equivalent to an effective divisor.

Consider the exact sequence

0 ⟶ H^{0}(X, 𝒪(nD − A_{1})) ⟶ H^{0}(X, 𝒪(nD)) ⟶ H^{0}(A_{1}, 𝒪(nD)|_{A_{1}}) .

The rank of the middle term grows like n^{dim X} for sufficiently large and divisible n, but the rank of the term on the right grows at most like n^{dim A_{1}} = n^{dim X−1} . Therefore H^{0}(X, 𝒪(nD − A_{1})) must have positive rank for some sufficiently large and divisible n. This gives us (a).

The forward implication of part (b) follows immediately from part (a) (and in fact the ample divisor can be chosen to be any given ample divisor). The converse follows from the fact that ample divisors are big, and adding an effective divisor does not decrease the dimension of the divisor. □

§9. Descent ——— Write this. Should it go earlier? See Serre, Groupes Alg. et Corps de Classes, p. 108 (Ch. V No. 20).

References

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Department of Mathematics, University of California, Berkeley, CA 94720