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�nally, div(x1 − x2) = Δ + Σ − 2(o × X + X × o). This proves the lemma. Proof of Theorem 3.13 Consider the map f : X → X × X given by f (α) =. Obviously p1 ◦ f = λ and p2 ◦ f = μ, so that f ∗(p∗ 1(o)) = λ∗(o) λ(α), μ(α) and f ∗(p∗ 2(o)) = μ∗(o) for λ, μ = 0. Similarly, ψ ◦ f = λ + μ, so that, since 3 The Plane Cubic 179 Σ = ψ ∗(o), we get f ∗(Σ) = (λ + μ)∗(o) for λ + μ = 0, and similarly f ∗(Δ) = (λ − μ)∗(o) for λ − μ = 0. Therefore applying f ∗ to (3.44) gives λ∗(o) + μ∗(o) (λ + μ)∗(o) + (λ − μ)∗(o) ∼ 2 (linear equivalence of divisor on X), provided that λ, μ, λ + μ, λ − μ = 0. Since linearly equivalent divisors have the same degree, and deg λ∗(o) = deg λ = n(λ), and similarly for μ, λ + μ, λ − μ, (3.43) follows, provided that λ, μ, λ + μ, λ − μ = 0. If λ = 0 or μ = 0 then (3.43) is obvious. If, say, λ + μ = 0 then we need only use the assertion in Example of Section 1.3, together with n(λ + μ) = 0; similarly if λ − μ = 0. This proves (3.43) and the theorem. 3.4 Applications Example 3.4 (m-torsion points of X) Consider the homomorphism δm : X → X of multiplication by m in X, that is, δm(α) = α ⊕ · · · ⊕ α " % #$ m times
. By (3.43) with λ = μ = δ1 it follows that n(δ2) = 4, and, by induction on m, one sees that n(δm) = m2. Suppose that k has characteristic 0. By Theorem 2.29, there exists a nonempty open set U ⊂ X such that points α ∈ U have exactly m2 inverse images under δm. But δm is a group homomorphism, so that the number of inverse images of any point is equal to the order of the kernel. We deduce that the number of solutions in X of the equation mα = 0 is equal to m2. Suppose now that k has characteristic p > 0, but m is not divisible by p. In order to be able to apply Theorem 2.29, we need to prove that δm is separable. If this were not the case, then by Theorem 2.31, we could write δm in the form g ◦ ϕ, where ϕ is the Frobenius map of X, and then deg δm would be divisible by p, whereas we know that deg δm = m2 and p m. Thus the number of solutions of mα = 0 equals m2 provided that m is not divisible by char k. In particular, the equation 3α = 0 has 9 solutions if char k = 3. We saw in 3.2 that the points satisfying 3α = 0 are the inﬂexion points of X. Hence a nonsingular plane cubic has 9 inﬂexion points. These enjoy a number of remarkable properties. For example, the line through any two of them intersects X again in an inﬂexion point. This follows at once from the fact that the sum of two solutions of 3α = 0 in a group is again a solution. Example 3.5 (Hasse–Weil estimates) Suppose now that X ⊂ P2 is a nonsingular plane cubic curve deﬁned by an equation with coefﬁcients in the ﬁeld Fp with p elements. In Example 1.16, we deﬁned the Frobenius map ϕ : (α1,..., αn) → (αp n ) for afﬁne varieties deﬁned over F
p. This deﬁnition extends automatically to arbitrary quasiprojective varieties. By Theorem 2.31, deg ϕ = p. 1,..., αp 180 3 Divisors and Differential Forms We apply Theorem 3.13 to maps λ : X → X of the form a + bϕ with a, b ∈ Z, given by a + bϕ : α → α ⊕ · · · ⊕ α % #$ a times By Theorem 3.13, we know that n(a + bϕ) = (a + bϕ, a + bϕ), and hence ⊕ ϕ(α) ⊕ · · · ⊕ ϕ(α) % #$ b times " ". n(a + bϕ) = a2n(1) + 2ab(1, ϕ) + b2n(ϕ) = a2 + 2ab(1, ϕ) + b2p. (3.45) By deﬁnition n(a + bϕ) ≥ 0 for all a and b, so that, viewed as a quadratic form in a, b, n(a + bϕ) has positive discriminant, and therefore ≤ (1, ϕ) √ p. (3.46) √ On the other hand, it follows from (3.45) that 2(1, ϕ) = n(1 − ϕ) − p − 1, and hence (3.46) gives n(1 − ϕ) − p − 1 ≤ 2 (1 − ϕ)∗(o) Moreover, n(1 − ϕ) = deg(1 − ϕ)∗(o), and Supp consists of points α ∈ X with (1 − ϕ)(α) = o, that is, α = ϕ(α). These are the points of X with coordinates in Fp. We prove that all these points appear in the divisor (1 − ϕ)∗(o) with multiplicity 1. As in the preceding example, referring to Theorem 2.29, it is enough to prove that the map 1 − ϕ is separable. For this, by Theorem 2.31, we need to prove that 1 − ϕ = μϕ for any
map μ : X → X. But it would follow from this that 1 = (1 + μ)ϕ, and this contradicts deg ϕ = p > 1. (3.47) p. Thus (3.47) can be rewritten \|N − p − 1\| ≤ 2 √ p, (3.47) where N is the number of points of X with coordinates in Fp (including the point at inﬁnity). In other words, the number N0 of solutions of the congruence satisﬁes the inequality y2 ≡ x3 + ax + b mod p \|N0 − p\| ≤ 2 √ p. (3.48) (3.49) This result has the following interpretation: for a given residue x modulo p, the congruence (3.48) has ⎧ ⎨ ⎩ no solutions 2 solutions if if x3+ax+b p x3+ax+b p = −1, = 1, 3 The Plane Cubic where a the estimate p is the Legendre symbol. Hence N0 − p = x3 + ax + b p p−1 x=0 181 p−1 x=0 x3+ax+b p, and (3.49) gives √ p. (3.50) ≤ 2 The estimates (3.49) and (3.50) have many applications in number theory. 3.5 Algebraically Nonclosed Field Suppose that the coefﬁcients a, b in the (3.38) belong to some ﬁeld k0, not necessarily algebraically closed. Write k for the algebraic closure of k0. The deﬁnition, or the explicit formulas for the group law on X, show that the points of X with coordinates in k0 form a subgroup, denoted by X(k0). Example 3.6 (The Mordell–Weil theorem) Suppose that k0 = Q is the rational number ﬁeld. In this case, the theorem known as the Mordell theorem asserts that the group X(Q) is ﬁnitely generated. In principle, this is a description of the set of rational solutions of (3.38) in ﬁnite terms, just as that provided in the case of conics by the parametrisation of Section 1.2, Chapter 1. Example 3
.7 (Divisors and rationality) In the more general situation, when X is a nonsingular projective algebraic curve deﬁned by equations with coefﬁcients in a ﬁeld k0, we can apply the preceding theory to the ﬁeld extension k0 ⊂ k. In what follows we assume that k0 is a perfect ﬁeld (just to simplify the arguments somewhat). With X we associate a function ﬁeld k0(X) ⊂ k(X) over k0 consisting of rational functions in the coordinates with coefﬁcients in k0. Suppose that D = nixi is a divisor with xi ∈ X such that the coordinates of the points xi are contained in an intermediate ﬁeld k1 with k0 ⊂ k1 ⊂ k; we can assume here that k0 ⊂ k1 is a Galois extension. An automorphism σ of the ﬁeld extension k0 ⊂ k1 applied to the coordinates of a point xi ∈ X(k1) obviously takes it into a point σ (xi) ∈ X(k1). If for every point xi, and every automorphism σ ∈ Gal(k1/k0), the conjugate points σ (xi) appears in D with the same multiplicity, then we say that the divisor D is rational over k0. This applies, in particular, to the divisor of a function f ∈ k0(x). We write Lk0(D) for the vector subspace of functions f ∈ k0(X) such that div f + D > 0. This is a vector space over k0. Set k0(D) = dimk0 Lk0(D). The automorphisms of k1 over k0 take functions of L(D) to one another and preserve the subspace Lk0 (D). They are not, however, linear maps: σ (αf ) = σ (α)σ (f ) for α ∈ k and f ∈ L(D); transformations of this type are said to be quasilinear. The so-called main theorem on quasilinear maps (Proposition A.5) asserts that L(D)
is generated over k by the k0-vector subspace Lk0(D) of invariant elements, that is L(D) = k · Lk0(D). In particular, k0 (D) = (D). (3.51) 182 3 Divisors and Differential Forms If divisors D and D are rational over k0 and linearly equivalent then there exists a function f ∈ k0(X) such that D −D = div f. To prove this one must apply the main theorem on quasilinear maps to the 1-dimensional space of functions g ∈ k1(X) for which div g = D − D. Example 3.8 (Zeta functions and the Weil Riemann hypothesis) We return to the cubic X. In Section 2.3, Chapter 1, we deﬁned the zeta functions ZX(t) and ζX(s) for an afﬁne variety deﬁned by equations with coefﬁcients in Fp. The deﬁnition obviously extends to arbitrary quasiprojective varieties. If X is a curve then the cycles deﬁned in Section 2.3, Chapter 1 are rational divisors over Fp, and as such, are obviously irreducible, that is, cannot be expressed as sums of other rational divisors. The Euler product (1.19) can then be rewritten, as in the case of the Riemann zeta function, in the form ZX(t) = t deg D, where D runs through all rational divisors over Fp. In other words, D≥0 ZX(t) = ant n, where an is the number of effective rational divisors D of degree n. We are now in a position to determine this number explicitly when X is a cubic curve. We ﬁrst ﬁnd the number of linear equivalence classes of rational divisors of degree n. We proved in Section 3.1 that if deg D = n then D ∼ x + (n − 1)o. From the fact that D is rational over Fp, it follows that x ∈ X(Fp). Indeed, if the coordinates of x are contained in a Galois extension k1 of k0 then σ (x) ∼ x for any automorphism of this extension
, and hence σ (x) = x. Thus the number of divisor classes of degree n equals N, where N is the number of points x ∈ X(Fp). Now we ﬁnd the number of divisors in a given class, that is, the number of divisors D ∼ D0 where D0 is given. They correspond to nonzero functions f ∈ LFp (D), considered up to constant factors in Fp. Thus the number of rational divisors with D ∼ D0 and D > 0 is equal to the number of points of Pk0 (D)−1, that is, (pk0 (D) − 1)/(p − 1). By (3.51) k0(D) = (D), and by Theorem 3.11, (D) = n. Therefore an = pn−1 p−1 N, and we get that ∞ ZX(t) = 1 + N n= pn − 1 p − 1 pt 1 − pt − t 1 − t = 1 + (N − p − 1)t + pt 2 (1 − t)(1 − pt). We see that the zeta function ZX(t) is a rational function of t. Moreover, the inequality (3.47) shows that the roots α1 and α2 of the quadratic polynomial 1 + 3 The Plane Cubic 183 (N − p − 1)t + pt 2 are complex conjugate algebraic integers. Since their product equals 1/p, we have \|αi\| = p−1/2. For the zeta function ζX(s) = ZX(p−s) this gives that the zeros β1 and β2 lie on the line "s = 1/2. We get in this way an analogue of the Riemann hypothesis. Analogous results hold for arbitrary nonsingular projective varieties, but their proofs are very much harder (see for example Hartshorne [37, Appendix C] for a survey). 3.6 Exercises to Section 3 1 Find all points of order 2 on a cubic curve in Weierstrass normal form. 2 Prove that if two cubic curves intersect in exactly 9 points then any cubic through 8 of these points also passes through the 9th. 3 Prove that the x-coordinates of the inﬂexion points of the
cubic curve (3.38) satisfy f (x) = x4 + 2ax2 + 4bx + a2. Prove that if a, b ∈ R then not all four of these points can be real. [Hint: Use the fact that f (x) = 4(x3 + ax + b).] Prove that a real cubic has one or 3 real inﬂexion point. In the latter case, they are collinear. 4 Prove that there are 4 tangent lines to a cubic X through every point of X. (We only count x ∈ TxX if x is an inﬂexion point.) 5 Prove that the points of tangency of the 4 tangent lines to a cubic X through a point a ∈ X lie on a conic tangent to X at a. 6 Prove that if two cubics X1 and X2 with equation y2 = x3 + aix + bi for i = 1 and 2 are isomorphic, then there exists an isomorphism that takes their points at inﬁnity to one another. 7 Under the assumptions of Exercise 6, prove that an isomorphism between X1 and X2 that takes their points at inﬁnity to one another is given by a linear map. 8 Under the assumptions of Exercises 6–7, prove that if b1, b2 = 0, then the two cubics X1 and X2 are isomorphic if and only if a3 = a3 1/b2 1 2/b2 2. 9 Prove that the zeta function ζX(s) associated with a cubic satisﬁes the functional equation ζX(1 − s) = ζX(s). 10 Prove that over a ﬁeld of characteristic p, any map of the cubic α : X → X with α(X) = X can be written in the form α = ϕr β where ϕ is the Frobenius map, r ≥ 0 and β is separable. 184 3 Divisors and Differential Forms 4 Algebraic Groups The results of the preceding sections lead to an interesting topic in algebraic geometry, the theory of algebraic groups. We will not go very deeply into this theory, but in order to give the reader at least an impression, we discuss some of its basic results, leaving out most
of the proofs. 4.1 Algebraic Groups The plane cubic curves of the preceding section are one of the most important examples of a general notion that we now introduce. Deﬁnition An algebraic group is an algebraic variety G which is at the same time a group, in such a way that the following conditions are satisﬁed: the maps ϕ : G → G given by ϕ(g) = g−1 and ψ : G × G → G given by ψ(g1, g2) = g1g2 are regular maps (here g−1 and g1g2 are the inverse and product in the group G). Examples of algebraic groups Example 3.9 A nonsingular plane cubic curve with the group law ⊕. Theorem 3.12 asserts that the conditions in the deﬁnition of algebraic group are satisﬁed. Example 3.10 The afﬁne line A1, with the group law deﬁned by addition of coordinates of points. This is called the additive group, and denoted by Ga. Example 3.11 The variety A1 \ 0, where 0 is the origin, with the group law deﬁned by multiplication of coordinates of points. This is called the multiplicative group, and denoted by Gm. Example 3.12 The open subset of the space An2 of n×n matrixes consisting of nondegenerate matrixes, with the usual matrix multiplication. This is called the general linear group. Example 3.13 The closed subset of the space An2 of n × n matrixes consisting of orthogonal matrixes, with the usual matrix multiplication. This is called the orthogonal group. We give a very simple example illustrating how being an algebraic group affects the geometry of the variety G. Theorem 3.14 The variety of an algebraic group is nonsingular. 4 Algebraic Groups 185 Proof It follows from the deﬁnition of an algebraic group that for any h ∈ G the map th : G → G given by th = hg is an automorphism of the variety G. For any g1, g2, we have th(g1) = g2, where −1 h = g2g 1, and the property that a point is singular is invariant under isomorphism, so that if any point of G
is singular, then so are all its points. But this contradicts the fact that the singular points of any algebraic variety form a proper closed subvariety. Therefore G does not have singular points. The theorem is proved. A generalisation of this situation is the case when a variety X has a group G of automorphisms, with the property that for any two points x1, x2 ∈ X there exists g ∈ G such that g(x1) = x2. In this case we say that X is homogeneous. The argument just given shows that a homogeneous variety is nonsingular. An example is the Grassmannian (Example 1.24). 4.2 Quotient Groups and Chevalley’s Theorem This section contains statements of some of the basic theorems on algebraic groups. Theorems are labelled with letters (Theorem A, etc.) to indicate that proofs are omitted.9 Deﬁnition An algebraic subgroup of an algebraic group G is a subgroup H ⊂ G that is a closed subset in G. As in the theory of abstract groups, a subgroup H ⊂ G is a normal subgroup if g−1Hg = H for every g ∈ G. Finally, a homomorphism ϕ : G1 → G2 of algebraic groups is a regular map that is a homomorphism of abstract groups. The problem of constructing the quotient group by a given normal subgroup is quite delicate. The difﬁculty, of course, is how to turn G/N into an algebraic variety. Theorem A The abstract group G/N can be made into an algebraic variety in such a way that the following conditions are satisﬁed: 1. The natural map ϕ : G → G/N is a homomorphism of algebraic groups. 2. For every homomorphism of algebraic groups ψ : G → G1 whose kernel contains N, there exists a homomorphism of algebraic groups f : G/N → G1 such that ψ = f ◦ ϕ. 9For a modern introduction to algebraic groups, Humphreys [40]. including proofs of Theorems A–B, see 186 3 Divisors and Differential Forms The algebraic group G/N is obviously uniquely determined by conditions 1 and 2. It is called the quotient group of G by N.
An algebraic group G is afﬁne if the algebraic variety G is afﬁne, and is an Abelian variety if G is projective and irreducible. The general linear group (Example 3.12) is obviously an afﬁne group. Indeed, it is the principal open set (Section 4.2, Chapter 1) of An2 deﬁned by det M = 0. Hence any algebraic subgroup of the general linear group is afﬁne. Theorem B An afﬁne algebraic group is isomorphic to an algebraic subgroup of the general linear group. Theorem C (Chevalley’s theorem) Every algebraic group G has a normal subgroup N such that N is an afﬁne group, and G/N an Abelian variety. The subgroup N is uniquely determined by these properties. 4.3 Abelian Varieties In the deﬁnition of Abelian variety, the projectivity condition on the algebraic group G contains a surprising amount of information; many unexpected properties of Abelian varieties ﬂow from it. We deduce here the simplest of these, which only require application of simple theorems already proved in Chapter 1. We need a property of arbitrary projective varieties. Deﬁne a family of maps X → Z between varieties to be a map f : X × Y → Z, where Y is some algebraic variety, the base of the family. Obviously for any y ∈ Y we have a map fy : X → Z deﬁned by fy(x) = f (x, y), which justiﬁes the terminology. Lemma Suppose that X and Y are irreducible varieties with X projective, and let f : X × Y → Z be a family of maps from X to a variety Z with base Y. Suppose that for some point y0 ∈ Y, the image f (X × y0) = z0 ∈ Z is a point. Then f (X × y) is a point for every y ∈ Y. Proof Consider the graph Γ of f. Obviously Γ ⊂ X × Y × Z and Γ is isomorphic to X × Y. Write for the projection to the second and third factors, and Γ = p(Γ ). Since X is projective, Γ is
closed by Theorem 1.11. Now let q : Γ → Y be the restriction of the ﬁrst projection Y × Z → Y. The ﬁbre of q over y is of the form y × fy(X × y), and so is nonempty, so that q(Γ ) = Y. On the other hand, by assumption, the ﬁbre over y0 consists of a single point y0 × z0. By the theorem on dimension of ﬁbres, Theorem 1.25, we see that dim Γ = dim Y. Choose any point x0 ∈ X. We obviously have {(y, f (x0, y)) \| y ∈ Y } ⊂ Γ, and this is a variety isomorphic to Y. Now since both of these varieties are irreducible and of the same dimension, they must be equal, and therefore f (X × y) = f (x0, y). The lemma is proved. 4 Algebraic Groups Figure 15 The map f (x, y) = xy 187 Remark Without the assumption that X is projective, the lemma is false, as shown by the family of maps fy : A1 → A1 given by f (x, y) = xy. The reason is that the set Γ is not closed, and Theorem 1.25 is not applicable to it. In the example Γ ⊂ A1 × A1 = A2 consists of all points (u, v) except those with u = 0, v = 0. That is, it is the plane with the line u = 0 deleted, but the origin u = v = 0 kept (Figure 15). Theorem 1.25 is really false for the projection q : (u, v) → u: the domain has dimension 2, the image dimension 1, but the ﬁbre over 0 has dimension 0. Theorem 3.15 An Abelian variety is an Abelian group. Proof Consider the family of maps from G to G with base G given by f (g, h) = g−1hg. Obviously when h = e is the identity element we have f (g, e) = e, and hence by the lemma, f (G, h) is a point for every h. Hence f (g, h) = f (e, h) = h, and
this means that G is Abelian. Theorem 3.16 If ψ : G → H is a regular map of an Abelian variety G to an algebraic group H, then ψ(g) = ψ(e)ϕ(g), where e ∈ G is the identity element, and ϕ : G → H a group homomorphism. Proof We set ϕ(g) = ψ −1(e)ψ(g) and prove that ϕ is a homomorphism. For this, consider the following family of maps G → H with base G: f : G × G → H given by f g, g g = ϕ ϕ(g)ϕ gg −1. Then since ϕ(e) = e is the identity element of H, we have f (G, e) = e. By the lemma, f (G, g) is a single point for every g ∈ G, that is, f (g, g) does not depend on g. Setting g = e we get f (g, g) = f (e, g) = e, and this means that ϕ is a homomorphism. Corollary If two Abelian varieties are isomorphic as algebraic varieties, they are isomorphic as groups; that is, “the geometry determines the algebra”. In particular, the maps of the cubic curve λ : X → X with λ(o) = o considered in Section 3.3 are homomorphisms. 188 3 Divisors and Differential Forms 4.4 The Picard Variety The only examples of Abelian varieties appearing so far are the plane cubic curves considered in Section 3. The group law on these was deﬁned starting out from the study of their divisor class groups. This example is typical of a much more general situation. Starting from an arbitrary nonsingular projective variety X, we can construct an Abelian variety whose group of points is isomorphic to a certain subgroup of the divisor class group Cl X, corresponding to Cl0 X in the case of the cubic curve. We give this deﬁnition, omitting proofs of all but the simplest assertions. We now deﬁne a new equivalence relation for divisors, algebraic equivalence. It is a coarser relation than the linear equivalence of divisors considered up to now (that is,
linear equivalence implies algebraic equivalence). Our aim is to study divisors on nonsingular varieties, but divisors on arbitrary varieties will appear at intermediate stages of the argument. In this case, we always take divisors to mean locally principal divisors. Let X and T be two arbitrary irreducible varieties. For any point t ∈ T the map jt : x → (x, t) deﬁnes an embedding X → X × T. Every divisor C on X × T with Supp C ⊃ X × t deﬁnes a pullback divisor j ∗ t (C) on X. In this case we say that j ∗ t (C) is deﬁned. Deﬁnition A family of divisors on X with base T is any map f : T → Div X. We say that the family f is an algebraic family of divisors if there exists a divisor C ∈ Div(X × T ) such that j ∗ t (C) = f (t). Divisors D1, D2 on X are algebraically equivalent if there exists an algebraic family of divisors f on X with base T, and two points t1, t2 ∈ T such that f (t1) = D1 and f (t2) = D2. This equivalence relation is denoted by D1 ≡ D2. t (C) is deﬁned for each t ∈ T and j ∗ Thus algebraic equivalence of divisors means that they can be “algebraically deformed” into one another. Algebraic equivalence is obviously reﬂexive and symmetric. It is easy to prove that it is transitive: if an algebraic equivalence between D1 and D2 is realised by a divisor C on X × T and an algebraic equivalence between D2 and D3 by a divisor C on X × T, then to prove that D1 and D3 are equivalent we need to consider the divisor (C × T ) + (C × T ) − D2 × T × T on X × T × T. We leave the detailed veriﬁcation to the reader. Finally, one sees easily that algebraic equivalence is compatible with addition in Div X: div
isors D with D ≡ 0 form a subgroup. We denote this by Diva X. Linear equivalence of divisors implies algebraic equivalence. It is enough to prove this for equivalence to 0. Suppose that D ∈ Div X and D ∼ 0, that is, D = div g with g ∈ k(X). Consider the variety T = A2 \ (0, 0), and write u, v for coordinates on A2. We can view g, u, v as functions on X × T, meaning the pullbacks p∗(g), q∗(u) and q∗(v), where as usual p : X × T → X and q : X × T → T are the projections. Set C = div(u + vg) and consider the algebraic family of divisors f deﬁned by the divisor C on X × T. One checks that f (1, 0) = 0 (the zero divisor) and f (0, 1) = D, and hence D ≡ 0. 4 Algebraic Groups 189 Finally, consider algebraic equivalence of divisors in the example of a nonsingular projective curve X. Then x ≡ y for any two points x, y ∈ X. For this, it is enough to consider the family of divisors f parametrised by X itself, and deﬁned by the diagonal Δ ⊂ X × X; it is easy to check that f (x) = x for every x ∈ X. Hence for every divisor D = nixi and any point x0 ∈ X we have D ≡ ( ni)x0, that is, any two divisors of the same degree are algebraically equivalent. It is slightly more complicated to prove the converse implication, that algebraically equivalent divisors have the same degree. We do not give the proof here.10 Thus for divisors on a nonsingular projective curve X, algebraic equivalence of divisors is equivalent to them having equal degree. Therefore Div X/ Diva X = Cl X/ Cl0 X = Z. A generalisation of this is the following theorem proved by Severi (for ﬁelds of characteristic 0) and Néron (in the general case). Theorem D (The Néron–Severi Theorem) For X a nons
ingular projective variety, the group NS X = Div X/ Diva X is ﬁnitely generated. One can show that algebraic and linear equivalence of divisors coincide on X = Pn1 × · · · × Pnk. This example shows that Div X/ Diva X can be more complicated that Z. When X is a plane cubic curve, the quotient Cl0 X = Diva X/P (X), where P (X) is the group of principal divisors, is a 1-dimensional Abelian variety. In a similar way, for any nonsingular projective variety X there exists an Abelian variety G whose group of points is isomorphic to Diva X/P (X), that is, divisors algebraically equivalent to 0 modulo divisors linearly equivalent to 0, and having the following property: for any algebraic family of divisors f on X over a base T there exists a regular map ϕ : T → G such that f (t) − f (t0) ∈ ϕ(t), where t0 is some ﬁxed point of T. (Here G is identiﬁed with Diva X/P (X), so that ϕ(t) is considered as a divisor class.) The Abelian variety G is uniquely determined by this property. It is called the Picard variety of X. The Picard variety of a nonsingular projective curve X is also called the Jacobian of X. 4.5 Exercises to Section 4 1 Let G be an algebraic group, ψ : G × G → G the regular map deﬁned by the group law, and Θe and Θe the tangent spaces to G and G × G at their respective 10The so-called “Principle of conservation of number” (roughly, algebraically equivalent cycles have the same numerical properties) is discussed in Fulton [29, Chapter 10], although the proof given there is very high-powered and abstract. Section 19.3 of the same book also contains a condensed discussion of the Néron–Severi theorem over C. 190 3 Divisors and Differential Forms identity elements. Prove that Θe = Θe ⊕ Θe and that deψ : Θe ⊕ Θe → Θe is given by addition of vectors
. 2 In the notation of Exercise 1, suppose that G is an Abelian group, and deﬁne ϕn : G → G by ϕn(g) = gn. Supposing that the ground ﬁeld has characteristic 0, prove that deϕn is a nondegenerate linear map. Deduce from this that in a Abelian algebraic group the number of elements of order n is ﬁnite, and that every element has an nth root. 5 Differential Forms 5.1 Regular Differential 1-Forms We introduced the notion of the differential dxf of a regular function f at a point x ∈ X of a variety in Section 1.3, Chapter 2. By deﬁnition, dxf is a linear form on the tangent space Θx to X at x, that is, dxf ∈ Θ ∗ x. We now study how this notion depends on x. Fix a function f regular everywhere on X. Then, as a function of x, the differential dxf is an object of a new type that we have not met before: it sends each point x ∈ X to a vector dxf ∈ Θ ∗ x in the dual space of the tangent space at x. Objects of this nature will appear all the time in what follows. Perhaps the following explanation will be helpful. In linear algebra, we deal with constants, but also with other quantities, such as vectors, linear forms, and arbitrary tensors. In geometry, the analogue of a constant is a function, which takes constant values at points. The analogue of a vector, linear form or whatever, is a “ﬁeld”, or “function” that sends each point x of an algebraic variety (or differentiable manifold) into a vector, linear form or whatever, of the tangent space Θx at the point x. Consider the set Φ[X] of all possible functions ϕ sending each point x ∈ X to a vector ϕ(x) ∈ Θ ∗ x. This set is of course much too big to be of any interest, just as the set of all k-valued function on X is too big. Now, just as we distinguished the regular functions among all k-valued functions on X, we now distinguish in Φ[X] a subset that is more closely related to the structure of X.
For this we note that Φ[X] is an Abelian group, if we set (ϕ + ψ)(x) = ϕ(x) + ψ(x). Moreover, Φ[X] is a module over the ring of all k-valued functions on X, if we set (f ϕ)(x) = f (x)ϕ(x) for f : X → k and ϕ ∈ Φ[X]. In particular we may view Φ[X] as a module over the ring k[X] of regular functions on X. As we have seen, a regular function f on X deﬁnes a differential dxf ∈ Θ ∗ x at x. Thus any function f ∈ k[X] deﬁnes an element ϕ ∈ Φ[X] by ϕ(x) = dxf. We denote this function by df. Deﬁnition An element ϕ ∈ Φ[X] is a regular differential form on X if every point x ∈ X has a neighbourhood U such that the restriction of ϕ to U belongs to the k[U ]-submodule of Φ[U ] generated by the elements df with f ∈ k[U ]. 5 Differential Forms 191 All the regular differential forms on X obviously form a module over k[X]; we denote it by Ω[X]. Thus ϕ ∈ Ω[X] if it can be written in the form ϕ = m i=1 fidgi, (3.52) in a neighbourhood of every point x ∈ X, where f1,..., fm, g1,..., gm are regular functions in a neighbourhood of x. Taking the differential of functions deﬁnes a map d : k[X] → Ω[X]. The proper- ties (2.6) then take the form d(f + g) = df + dg and d(fg) = f dg + gdf. (3.53) From these formulas one deduces easily an identity that holds for any polynomial F ∈ k[T1,..., Tm] and any functions f1,..., fm ∈ k[X]: F (f1,..., fm) d = m i=1 ∂F ∂
Ti (f1,..., fm)dfi. (3.54) To obtain this, using (3.53), one reduces the proof to the case of a monomial, and then, using (3.53) again, proves it by induction of the degree of the monomial. We leave the details of the veriﬁcation to the reader. Once (3.54) is proved for polynomials, it generalises immediately to rational functions F. Here we should note that if a rational function F is regular at x, then so are all the ∂F /∂Ti ; indeed, then F = P /Q, where P, Q are polynomials and Q(x) = 0, and so ∂F ∂Ti = 1 Q2 whence regularity. Q ∂P ∂Ti − P ∂Q ∂Ti, Example 3.14 (X = An) Since the differentials dxt1,..., dxtn of the coordinates form a basis of the vector space Θ ∗ x at any point x ∈ An, any element ϕ ∈ Φ[An] n can be written uniquely in the form ϕ = i=1 ψidti, where the ψi are k-valued functions on An. If ϕ ∈ Ω[An] then ϕ must have an expression (3.52) in a neighbourhood of any point x ∈ An. Applying the relations (3.54) to gi gives an expression ϕ = hidti, in which the hi are rational functions regular at x. Since such an expression is unique, the ψi must be regular at every x ∈ An, that is, ψi ∈ k[An]. Therefore An = Ω! An dti. k ∼= Example 3.15 Let X = P1 with coordinate t. Then X = A1 0 A1. By the result of Example 3.14, any element ϕ ∈ Ω[P1] can be written ϕ = 1, where A1 0 ∼= A1 1 ∪ A1 192 3 Divisors and Differential Forms P (t)dt on A1 du = −dt/t 2, so that if n = deg Q, we have in A1 0 0 and ϕ = Q(u)du on A1 ∩ A1
1 1, where u = t −1. The ﬁnal relation gives P (t)dt = − 1 t 2 Q(1/t)dt, that is, P (t) = −Q∗(t) t n+2, where Q∗(t) = t nQ(1/t), so that Q∗(0) = 0. A relation of this kind between polynomials is only possible if P = Q = 0, and hence Ω[P1] = 0. Example 3.16 Suppose that X ⊂ P2 is given by the equation x3 = 0, 0 and that char k = 3. Write Uij for the open set in which xi, xj = 0. Then X = U01 ∪ U12 ∪ U20. We set + x3 2 + x3 1 x = x1 x0 u = x2 x1 s = x0 x2,,, y = x2 x0 v = x0 x1 t = x1 x2 and ϕ = dy x2 and ψ = dv u2 and χ = dt s2 in U01; in U12; in U20. Obviously ϕ ∈ Ω[U01], ψ ∈ Ω[U12] and χ ∈ Ω[U20]. It is easy to check that ϕ = ψ = χ in U01 ∩ U12 = U01 ∩ U20 = U12 ∩ U20. Hence ϕ, ψ and χ deﬁne a global form ω ∈ Ω[X]. This example is interesting, in that Ω[X] = 0, whereas X is a projective variety, and so has no everywhere regular functions other than the constants. In the general case one can prove a weaker version of the result of Example 3.14. Theorem 3.17 Any nonsingular point x of an algebraic variety X has an afﬁne neighbourhood U such that Ω[U ] is a free k[U ]-module of rank dimx X. Proof Let X ⊂ AN, and suppose that F1,..., Fm are polynomials forming a basis of the ideal AX. Then Fi = 0 on X, and hence by (3.54) we have N j =1 ∂
Fi ∂Tj dtj = 0. (3.55) If x ∈ X is a nonsingular point and dimx X = n then the matrix (∂Fi/∂Tj )(x) has rank N − n. Suppose, for example, that t1,..., tn are local parameters at x. Then it follows from (3.55) that all the dtj can be expressed in terms of dt1,..., dtn with coefﬁcients that are rational functions, regular at x. Consider a neighbourhood U of x in which all these functions are regular. Then y for every y ∈ U. Let ϕ ∈ Ω[U ]. By what we have dyt1,..., dytn form a basis of Θ ∗ 5 Differential Forms said above, there exists a unique expression ϕ = n i=1 ψidti, 193 (3.56) where ψi are k-valued functions on U. By the expression (3.52) and formula (3.54) it follows that ϕ can be written in a neighbourhood of any point y ∈ U as a linear combination of dt1,..., dtN with coefﬁcients functions that are regular at y. As we have seen, dt1,..., dtN can in turn be expressed in the same way in terms of n dt1,..., dtn. Hence ϕ = i=1 gidti, where the gi are regular in a neighbourhood of y. Since the expression (3.56) is unique, it follows that ψi = gi in a neighbourhood of each y ∈ U, and hence ψi ∈ k[U ]. We see that Ω[U ] = n i=1 gidti = 0, with, say, gn = 0. Then dt1,..., dtn are linearly dependent in the open set gn = 0, which contradicts the linear independence of dyti ∈ Θ ∗ n i=1 k[U ]dti. The theorem is proved. y for all y ∈ U. Therefore Ω[U ] = Suppose that dt1,..., dtn are related by n i=1 k[U ]dti. ) Coroll
ary If u1,..., un is any system of local parameters at a point x, then du1,..., dun generate Ω[U ] as a k[U ]-module for some afﬁne neighbourhood U of x. Proof Let dt1,..., dtn be the basis of the free module Ω[U ] in a neighbourhood U n of x that exists by Theorem 3.17. Then dui = j =1 gij dtj, and since the ui are local parameters, det \|gij (x)\| = 0. Therefore in the neighbourhood U where det \|gij \| = 0, the elements du1,..., dun generate Ω[U ]. 5.2 Algebraic Deﬁnition of the Module of Differentials We saw in Chapter 1 that the category of afﬁne varieties is equivalent to the category of rings of a special type. Thus the whole theory of afﬁne varieties can be seen from a purely algebraic point of view; in particular, we can try to understand the algebraic meaning of the module of differential forms. Consider an afﬁne variety X, and write A = k[X] for its coordinate ring and Ω = Ω[X] for the module of differentials. Taking the differential of a function deﬁnes a k-linear homomorphism d : A → Ω. Proposition 3.1 Ω is generated as an A-module by the elements df with f ∈ A. Proof This is analogous to Theorem 1.7, and the proof is entirely similar. If ω ∈ Ω fi,xdgi,x with fi,x, gi,x ∈ Ox. then, by deﬁnition, for any x ∈ X we can write ω = Every function u ∈ Ox can be expressed as u = v/w with v, w ∈ A and w(x) = 0. 194 3 Divisors and Differential Forms Using such expressions for fi,x, gi,x and taking a common denominator for all the fractions, we obtain a function px such that px(x) = 0 and pxω = ri,xdhi,x with ri,x, hi,x ∈ A. Now because px(x) = 0 for each x
∈ X, there exist a ﬁnite set of x and functions qx ∈ A such that i qxri,xdhi,x. This proves Proposition 3.1. pxqx = 1. Therefore ω = x Proposition 3.1 suggests the idea of trying to describe Ω in terms of its generators df with f ∈ A. The following relations obviously hold: d(f + g) = df + dg, d(fg) = f dg + gdg, and dα = 0 for α ∈ k. (3.57) Proposition 3.2 If X is a nonsingular afﬁne variety and A = k[X] then the Amodule Ω is deﬁned by the relations (3.57). Proof Write R for the A-module having generators df in one-to-one correspondence with elements f ∈ A, and relations (3.57). There is an obvious homomorphism ξ : R → Ω, and Proposition 3.1 shows that ξ is surjective. It remains to prove that ker ξ = 0. Suppose that ϕ ∈ R and ξ(ϕ) = 0. We observe that the arguments in the proof of Theorem 3.17 only used the relations (3.57). Hence they are applicable also to R and show that for any x ∈ X there exists a function p ∈ A such that p(x) = 0 and pϕ = gidti with gi ∈ A, where now the local parameters ti are chosen as elements of A. If ξ(ϕ) = 0, then gidti = 0 in the module Ω, and it follows from Theorem 3.17 that all the gi = 0. Thus pϕ = 0. We see that for every x ∈ X there exists a function p ∈ A such that p(x) = 0 and pϕ = 0. Arguing as in the proof of Proposition 3.1, we see that ϕ = 0. The proposition is proved. Remark It follows easily from what we have just said that for any A-module M we have HomA(ΩA/A0, M) = DerA0 (A, M), where DerA0(A, M) is the module of derivations of A to M, that is, of A
0-linear maps D : A → M satisfying D(ab) = aD(b) + bD(a) for all a, b ∈ A. Thus in this case Ω[X] can be described purely algebraically starting from the ring k[X]. This suggests the idea of considering a similar module for any ring A, with A an algebra over a subring A0. The module deﬁned by generators df for f ∈ A and relations (3.57) (where of course α ∈ A0 in the ﬁnal relation) is called the module of differentials or module of Kähler differentials of A over A0 and denoted by ΩA or ΩA/A0. 5 Differential Forms 195 If X is singular, then the module ΩA deﬁned in this purely algebraically way no longer coincides with Ω[X] in general (see Exercise 9). Proposition 3.1, which still holds for a singular variety X, shows that ΩA contains more information on X than Ω[X]. However, in what follows, we mostly deal with nonsingular varieties, and the distinction will not be important for us. 5.3 Differential p-Forms The differential forms considered in Section 5.1 were functions sending each point x ∈ X to an element of Θ ∗ x. We now consider more general differential forms, that send x ∈ X to a skewsymmetric multilinear form on Θx, that is, to an element of the rth exterior power r Θ ∗ x of Θ ∗ x. The deﬁnition is entirely analogous to that considered in Section 5.1. We write Φr [X] for the set of all possible functions sending each point x ∈ X to an element x. In particular, Φ0[X] of is the ring of all k-valued functions on X and Φ1[X] is the set Φ[X] considered in the preceding section. Hence df ∈ Φ1[X] for f ∈ k[X]. x. Thus if ω ∈ Φr [X] and x ∈ X then ω(x and ψ ∈ We recall the operation of exterior product ∧, which is deﬁned for any vector r+s L. Moreover, the
exterior s L, then ϕ ∧ ψ ∈ space L. If ϕ ∈ product is distributive, associative and satisﬁes ϕ ∧ ψ = (−1)rsψ ∧ ϕ. If e1,..., en ∧ · · · ∧ eir is a basis of L then a basis of with 1 ≤ i1 < · · · < ir ≤ n. Hence dim (the binomial coefﬁcient), and in particular dim n r r L = 0 for r > n. r L is given by all the products ei1 n L = 1 and r L = We deﬁne the exterior product on the sets Φr [X]. If ωr ∈ Φr [X] and ωs ∈ Φs[X] then we deﬁne ω = ωr ∧ ωs by ω(x) = ωr (x) ∧ ωs(x) for all x ∈ X. Obviously ω ∈ Φr+s[X]. For r = 1 and s = 0 we return to the multiplication of elements of Φ1[X] = Φ[X] by k-valued functions on X. Taking any r and s = 0, we see that elements of Φr [X] can be multiplied by functions on X. In particular all the Φr [X] are modules over k[X]. Deﬁnition An element ϕ ∈ Φr [X] is a regular differential r-form on X if any point x ∈ X has a neighbourhood U such that ϕ on U belongs to the submodule of Φr [U ] generated over k[U ] by the elements df1 ∧ · · · ∧ dfr with f1,..., fr ∈ k[U ]. In terms of this deﬁnition, the differential forms considered in Sections 5.1–5.2 are regular differential 1-forms. All regular differential r-forms on X form a k[X]-module, denoted by Ω r [X]. Thus an element ω ∈ Ω r [X] can be written in a neighbourhood of any point x ∈ X in the form ω = gi1...ir dfi1 ∧
· · · ∧ dfir, (3.58) where gi1...ir and fi1,..., fir are regular functions in a neighbourhood of x. The exterior product is deﬁned on regular differential forms, and for ωr ∈ Ω r [X] and ωs ∈ Ω s[X], we obviously have ωr ∧ ωs ∈ Ω r+s[X]. 196 3 Divisors and Differential Forms Theorem 3.17 has an analogue for the forms Ω r [X] for any r. Theorem 3.18 Any nonsingular point x ∈ X of an n-dimensional variety has a neighbourhood U such that Ω r [U ] is a free k[U ]-module of rank. n r Proof In the proof of Theorem 3.17, we saw that a nonsingular point x has a neighbourhood U on which there are n regular functions u1,..., un such that dyu1,..., dyun form a basis of Θ ∗ y for any y ∈ U. It follows from this that any element ϕ ∈ Φr [U ] is of the form ϕ = ψi1...ir dui1 ∧ · · · ∧ duir, where the ψi1...ir are k-valued functions on U. If ϕ ∈ Ω r [U ] then ϕ can be expressed in the form (3.57) in a neighbourhood of any point y ∈ U. Applying Theorem 3.17 to the forms dfi we see that ψi1...ir are regular at y; but since y is any point of U, they are regular functions on U. Thus the ∧ · · · ∧ duir for 1 ≤ i1 < · · · < ir ≤ n generate Ω r [U ]. It remains to see forms dui1 that they are linearly independent over k[U ]. But any dependence relation gi1...ir dui1 ∧ · · · ∧ duir = 0 gives a relation gi1...ir (x)dxui1 = 0 at any point x ∈ U. Since dxu1,..., dxun form a basis of Θ ∗ r Θ ∗ · · · ∧ dxuir form a basis of for
all x ∈ U, that is, gi1...ir ∧ x. Hence from (3.59) it follows that gi1...ir (x) = 0 = 0. The theorem is proved. x, the elements dxui1 ∧ · · · ∧ dxuir (3.59) Of special importance is the module Ω n[U ], which under the assumptions of Theorem 3.18 is of rank 1 over k[U ]. 11 Thus if ω ∈ Ω n[U ], we have ω = gdu1 ∧ · · · ∧ dun with g ∈ k[U ]. (3.60) This expression for ω depends in an essential way on the choice of the local parameters u1,..., un. We determine what this dependence is. Let v1,..., vn be another n regular functions on X such that v1 − v1(x),..., vn − vn(x) are local parameters at any point x ∈ U. Then also Ω 1[U ] = k[U ]dv1 ⊕ · · · ⊕ k[U ]dvn. and in particular the dui can be expressed dui = n j =1 hij dvj for i = 1,..., n. (3.61) 11Elements of Ω n[U ] are called canonical differentials, following a suggestion of Mumford. 5 Differential Forms 197 Since dxu1,..., dxun form a basis of the vector space Θ ∗ x for each x ∈ U, it follows from (3.61) that det \|hij (x)\| = 0. By analogy with what happens in analysis, det \|hij (x)\| is called the Jacobian determinant of the functions u1,..., un with respect to v1,..., vn. We denote it by J u1,...,un v1,...,vn ∈ k[U ], and J (x) = 0 (3.62). As we have seen, u1,..., un v1,..., vn u1,..., un v1,..., vn J for all x ∈ U. Substituting (3.61) in the expression for �
� and simple calculations in the exterior algebra shows that ω = gJ u1,..., un v1,..., vn dv1 ∧ · · · ∧ dvn. (3.63) Thus although ω ∈ Ω n[U ] is speciﬁed by a function g ∈ k[X], this speciﬁcation is only possible once local coordinates have been chosen, and depends in an essential way on this choice. We recall that the expression (3.60) is in general only possible locally (see the statements of Theorems 3.17–3.18). If X = Ui is an open cover, and in each Ui an expression (3.60) is possible, we still cannot associate with ω a global function g on the whole of X: the functions gi obtained on different Ui are not compatible. We have already seen an example of this in Example 3.15. 5.4 Rational Differential Forms Example 3.15 shows that there may be very few regular differential forms on an algebraic variety X (for example, Ω 1[P1] = 0) whereas there are lots on its open subsets (for example, Ω 1[U ] = k[u]du). A similar thing happened in connection with regular functions, and it was precisely these considerations that led us to introduce the notion of rational functions, as functions regular on some open subset. We now introduce the analogous notion for differential forms. Consider a nonsingular irreducible quasiprojective variety X. Let ω be a differential r-form on X. Recall that it makes sense to speak of ω being 0 at a point x ∈ X; for ω(x) ∈ x, and in particular, it can be 0 there. r Θ ∗ Lemma The set of points at which a regular differential form ω is 0 is closed. Proof Since closed is a local property, we can restrict ourselves to a sufﬁciently small neighbourhood U of any point x ∈ X. In particular we can choose U so that Theorems 3.17 and 3.18 hold for it. Then there exist functions u1,..., un ∈ k[U ] such that Ω r [U ] is the free k[U ]-module based by dui1 ∧ ·
· · ∧ duir for 1 ≤ i1 < · · · < ir ≤ n. 198 Hence ω has a unique expression in the form ω = the conditions ω(x) = 0 is equivalent to gi1...ir lemma is proved. gi1...ir dui1 ∧ · · · ∧ duir, and = 0, which deﬁne a closed set. The 3 Divisors and Differential Forms It follows in particular from the lemma that if ω(x) = 0 for all points x of an open set U then ω = 0 on the whole of X. We now introduce a new object, consisting of an open set U ⊂ X and a differential r-form ω ∈ Ω r [U ]. On pairs (ω, U ) we introduce the equivalence relation (ω, U ) ∼ (ω, U ) if ω = ω on U ∩ U. By the above remark, it is enough to require that ω = ω on some open subset of U ∩ U. The transitivity of the equivalence relation follows from this. An equivalence class under this relation is called a rational differential r-form on X. The set of all rational differential r-forms on X is denoted by Ω r (X). Obviously Ω 0(X) = k(X). Algebraic operations on representatives of equivalence classes carry over to the classes, and deﬁne the exterior product: if ωr ∈ Ω r (X) and ωs ∈ Ω s(X) then ωr ∧ ωs ∈ Ω r+s(X). When s = 0 we see that Ω r (X) is a k(X)-module. If a rational differential form ω (an equivalence class of pairs) contains a pair (ω, U ) then we say that ω is regular in U. The union of all open sets on which ω is regular is an open set Uω, called the domain of regularity of ω. Obviously ω deﬁnes a regular form belonging to Ω r [Uω]. If x ∈ Uω then we say that ω is regular at x. Obviously Ω r (X) does not change if we replace X by an open subset, that is, it is a birational invariant
. We determine the structure of Ω r (X) as a module over the ﬁeld k(X). Theorem 3.19 Ω r (X) is a vector space over k(X) of dimension. n r Proof Consider any open set U ⊂ X for which Ω r [U ] is free over k[U ], as in Theorems 3.17–3.18. Then there exist n functions u1,..., un ∈ k[U ] such that the products dui1 ∧ · · · ∧ duir for 1 ≤ i1 < · · · < ir ≤ n (3.64) form a basis of Ω r [U ] over k[U ]. Any form ω ∈ Ω r (X) is regular in some open subset U ⊂ U, over which (3.64) still gives a basis of Ω r [U ] over k[U ]. Hence ω can be uniquely written in the form 1≤i1<···<ir ≤n gi1...ir dui1 ∧ · · · ∧ duir, where gi1...ir are regular in some open set U ⊂ U, that is, are rational functions on X. This just means that the forms (3.64) are a basis of Ω r (X) over k(X). The theorem is proved. For which n-tuples of functions u1,..., un ∈ k(X) is dui1 ∧ · · · ∧ duir for 1 ≤ i1 < · · · < ir ≤ n a basis of Ω r (X) over k(X)? We give a sufﬁcient condition for this—in fact it is also necessary, but we do not need this. 5 Differential Forms 199 Theorem 3.20 If u1,..., un is a separable transcendence basis of k(X) then the ∧ · · · ∧ duir for 1 ≤ i1 < · · · < ir ≤ n form a basis of Ω r (X) over k(X). forms dui1 Proof Since Ω r (X) and k(X) are birational invariants, we can assume that X is afﬁne, X ⊂ AN. Let u1,..., un
be a separable transcendence basis of k(X). Then any element v ∈ k(X) satisﬁes a relation F (v, u1,..., un) = 0 that is separable in v. In particular for each of the coordinates ti of AN, there is a relation Fi(ti, u1,..., un) = 0, for i = 1,..., N. It follows from these that the relations ∂Fi ∂ti dti + n j =1 ∂Fi ∂uj duj = 0 for i = 1,..., N hold on X. Since Fi is separable in ti it follows that ∂Fi/∂ti = 0 on X. Hence dti = n j =1 − (∂Fi/∂uj ) (∂Fi/∂ti) duj. (3.65) All the function (∂Fi/∂uj )/(∂Fi/∂ti) and ui are regular on some open set U ⊂ X, and then (3.65) shows that at any point y ∈ U, the differentials dyuj generate Θ ∗ y. Since the number of these differentials is equal to dim X = dim Θ ∗ y, they form a basis. Hence the dui form a basis of Ω 1[U ] as a k[U ]-module, and the products (3.64) a basis of Ω r [U ] over k[U ], hence a fortiori of Ω r (X) over k(X). The theorem is proved. 5.5 Exercises to Section 5 1 Prove that the rational differential form dx/y is regular on the afﬁne circle X deﬁned by x2 + y2 = 1. We suppose that the ground ﬁeld has characteristic =2. 2 In the notation of Exercise 1, prove that Ω 1[X] = k[X](dx/y). [Hint: Write any form ω ∈ Ω 1[X] in the form ω = f dx/y, and use the fact that dx/y = −dy/x.] 3 Prove that dim Ω 1[X] = 1 in Example 3.16. 4 Prove that Ω n[Pn] = 0
. 5 Prove that Ω 1[Pn] = 0. 6 Prove that Ω r [Pn] = 0 for any r > 0. 7 Let ω = (P (t)/Q(t))dt be a rational form on P1, with coordinate t, where P and Q are polynomials with deg P = m, deg Q = n. At what points x ∈ P1 is ω not regular? 200 3 Divisors and Differential Forms 8 Prove that for a nonsingular variety X, the tangent ﬁbre space introduced in Section 1.4, Chapter 2 is birational to the product X × An. [Hint: For the open set U in Theorem 3.17, construct an isomorphism of the tangent ﬁbre space of U to U × An by (x, ξ ) → x, (dxu1)(ξ ),..., (dxun)(ξ ), for ξ ∈ Θx.] 9 Compute the module R = ΩA constructed in the proof of Proposition 3.2 when X is the curve y2 = x3, and prove that 3ydx − 2xdy is a nonzero element of ΩA, but ξ(3ydx − 2xdy) = 0 ∈ Ω 1[X] (where ξ : R = ΩA → Ω[X] is as in the proof of Proposition 3.2). Show also that y(3ydx − 2xdy) = x2(3ydx − 2xdy) = 0. [Hint: Use the fact that k[X] = k[x] + k[x]y. The point is that on a singular variety Kähler differentials (Section 5.2) and regular differentials are different notions.] 10 Let K be an extension ﬁeld of k. A derivation of K over k is a k-linear map D : K → K satisfying the conditions D(xy) = yD(x) + xD(y) for x, y ∈ K. Prove that if u ∈ K and D is a derivation, then the map D1(x) = uD(x) is also a derivations, so that all the derivations of K over k form a vector space over k. This is denoted
by Derk(K). 11 Let D be a derivation of K = k(X) over k, and ω = that the function (D, ω) = the form Derk(K) ∼= (Ω 1(X))∗ = Homk(X)(Ω 1(X), k(X)). fidgi ∈ Ω 1(X). Prove fiD(gi) is independent of the representation of ω in fidgi. Prove that it is a scalar product, and establishes an isomorphism 12 Prove that for any ring A that is a vector space over k, the module of differentials ΩA constructed in Section 5.2 has the property Homk(ΩA, B) ∼= Derk(A, B) for any A-module B. (For the deﬁnition of Derk(A, B), see Exercise 24 of Section 1.6, Chapter 2.) 6 Examples and Applications of Differential Forms 6.1 Behaviour Under Maps We ﬁrst study the behaviour of differential forms under regular maps. If ϕ : X → Y is a regular map, x ∈ X and y = ϕ(x) ∈ Y then dxϕ is a map dxϕ : ΘX,x → ΘY,y, and its dual a map (dxϕ)∗ : Θ ∗ X,x. Hence for ω ∈ Φ[Y ] we have a pullback Y,y ϕ∗(ω) ∈ Φ[X] deﬁned by ϕ∗(ω)(x) = (dxϕ)∗(ω(y)). It follows easily from the deﬁnition that (dxϕ)∗ is compatible with taking the differential, that is, (dxϕ)∗(dyf ) = dx(ϕ∗(f )) for f ∈ k[Y ]. It follows that if ω ∈ Ω 1[Y ] then ϕ∗(ω) ∈ Ω 1[X], and ϕ∗ deﬁnes a homomorphism ϕ∗ : Ω 1[Y ] → Ω 1[X] that is compatible with taking the differential of f ∈ k[Y ]. → Θ ∗ 6 Examples and Applications of
Differential Forms 201 Finally, it is known from linear algebra that a linear map ϕ : L → M between r M. Applying this to X,x, hence maps Φr [Y ] → vector spaces determines a linear map (dxϕ)∗, we get a map Φr [X] and Ω r [Y ] → Ω r [X]. These maps will also be denoted ϕ∗. r L → r Θ ∗ r (dxϕ)∗ : r ϕ : → r Θ ∗ Y,y From what we have said above, it follows that the effective computation of the action of ϕ∗ on differential forms is very simple: if ω = gi1...ir dui1 ∧ · · · ∧ duir, then. (3.66) ∧ · · · ∧ d ϕ∗(ω) = ϕ∗(gi1...ir )d ϕ∗(ui1) Now suppose that X is irreducible, and ϕ : X → Y a rational map such that ϕ(X) is dense in Y. Since ϕ is a regular map of an open set U ⊂ X to Y, and any open set V ⊂ Y intersects ϕ(U ), the preceding arguments deﬁne a map ϕ∗ : Ω r (Y ) → Ω r (X). This is again given by (3.66). ϕ∗(uir ) We know that for r = 0, in other words, for functions, the map ϕ∗ : k(Y ) → k(X) is an inclusion. For differential forms this is not always so. Suppose, for example, that X = Y = P1, with respective coordinates t and u, so that k(X) = k(t) and k(Y ) = k(u). Suppose that k has ﬁnite characteristic p and that ϕ is given by the formula u = t p. Then ϕ∗(f (u)) = f (t p), and ϕ∗(df ) = d(f (t p)) = 0 for all f ∈ k(u) (because dt p = pt p−1dt = 0), so that ϕ∗(Ω 1(Y )) =
0. The situation is clariﬁed by the following result. Theorem 3.21 If k(X) has a separable transcendence basis over k(Y ) then ϕ∗ : Ω r (Y ) → Ω r (X) is an inclusion. Here we identify k(Y ) with the subﬁeld ϕ∗(k(Y )) ⊂ k(X). Proof Suppose that the extension k(Y ) ⊂ k(X) has a separable transcendence basis v1,..., vs. This means that v1,..., vs are algebraically independent over k(Y ), and k(X) is a ﬁnite separable extension of the subﬁeld k(Y )(v1,..., vs). The ﬁeld k(Y ) has a separable transcendence basis over k (see Theorem 1.8, Remark 1.1). Denote this by u1,..., ut. Then u1,..., ut, v1,..., vs is a separable transcendence basis of k(X) over k. We write any differential form ω ∈ Ω r (Y ) in the form ω = gi1...ir dui1 ∧ · · · ∧ duir, (3.67) and apply (3.66) to it, giving an expression for ϕ∗(ω) as a linear combination of elements dϕ∗(ui1)∧· · ·∧dϕ∗(uir ), which, by the Néron–Severi Theorem (Theorem D), are a subset of a basis of Ω r (X) over k(X), since the ϕ∗(ui) are part of the separable transcendence basis u1,..., ut, v1,..., vs. Hence ϕ∗(ω) = 0 only if all ϕ∗(gi1...ir ) = = 0, that is ω = 0. The theorem is proved. 0, and this is only possible if all gi1...ir So far everything has been more or less obvious. We now arrive at an unexpected fact. 202 3 Divisors and Differential Forms Theorem 3
.22 If X and Y are nonsingular varieties, with Y projective, and ϕ : X → Y a rational map such that ϕ(X) is dense in Y, then ϕ∗Ω r [Y ] ⊂ Ω r [X]. In other words, ϕ∗ takes regular differential forms to regular differential forms. Since ϕ is only a rational map, this seems quite implausible, even for functions, that is, the case r = 0. In this case we are saved by the fact that, since Y is projective, the only regular functions on Y are constant, and the theorem is vacuous. In the general case, the theorem is less obvious. Proof We use the fact that by Theorem 2.12, ϕ is regular on X \ Z, where Z ⊂ X is a closed subset and codimX Z ≥ 2. If ω ∈ Ω r [Y ] then ϕ∗(ω) is regular on X \ Z. Let us prove that regularity on the whole of X follows from this. For this, we write ϕ∗(ω) in some open set U in the form ϕ∗(ω) = gi1...ir dui1 ∧ · · · ∧ duir, ∧ · · · ∧ duir is a basis where u1,..., un are regular functions on U such that dui1 for Ω r [U ] over k[U ]. Then from the fact that ϕ∗(ω) is regular on X \ Z it follows that all the functions gi1...ir are regular on U \ (U ∩ Z). But codimU (U ∩ Z) ≥ 2, and this means that the set of points where gi1...ir is not regular has codimension ≥2. On the other hand, this set is a divisor div∞(gi1...ir ). This is only possible if div∞(gi1...ir ) = 0, and hence gi1...ir are regular functions. The theorem is proved. Corollary If two nonsingular projective varieties X and Y are birational then the vector space Ω r [X] and Ω r [Y ] are isomorphic. The signiﬁcance of Theorem 3.22 and its corollary is enhanced by the fact
that for projective varieties X, the vector spaces Ω r [X] are ﬁnite dimensional over k. This result is a consequence of a general theorem on coherent sheaves proved in Theorem of Section 3.4, Chapter 6. For the case of curves we prove it in Section 6.3. Set hr = dim Ω r [X]. The corollary means that the numbers hr for r = 0,..., n are birational invariants of a nonsingular projective variety X. 6.2 Invariant Differential Forms on a Group Let X be an algebraic variety, ω a differential form on X, and g an automorphism of X. We say that ω is invariant under g if g∗(ω) = ω. Suppose in particular that G is an algebraic group (see Section 4.1 for the deﬁnition). It follows at once from the deﬁnition that for any element g ∈ G, the translation map tg : G → G given by tg(x) = gx 6 Examples and Applications of Differential Forms 203 is regular and is an automorphism of G as an algebraic variety. A differential form on G is invariant if it is invariant under all the translations tg. An invariant rational differential form is regular. Indeed, if ω is regular at a point g (ω) = ω, so that ω is regular at all x0 ∈ G then t ∗ points gx0 for g ∈ G, and these are all the points of G. g (ω) is regular at g−1x0. But t ∗ We show how to ﬁnd all invariant differential forms on an algebraic group. For this, consider the vector spaces Φr [G] as in Sections 5.1 and 5.3, and their automorphisms t ∗ g corresponding to the translations tg. We determine ﬁrst the set of elements ϕ ∈ Φr [G] that are invariant under all t ∗ g for g ∈ G. This set contains in particular all the invariant regular differential r-forms. The condition t ∗ g (ϕ) = ϕ means that for any point x ∈ G, r ϕ(x) =. ϕ(gx) dt ∗ g
In particular, for g = x−1, r ϕ(e) dt ∗ x−1 = ϕ(x). (3.68) (3.69) This formula shows that ϕ is uniquely determined by the element ϕ(e) of the ﬁnite dimensional vector space e. Conversely, if we specify an arbitrary element η ∈ e, we can use (3.69) to construct the element ϕ ∈ Φr [G] given by r Θ ∗ r Θ ∗ r dt ∗ x−1 (η). ϕ(x) = A simple substitution shows that it also satisﬁes (3.68), that is, is invariant under t ∗ g. Thus the subspace of elements of ϕ ∈ Φr [G] that are invariant under t ∗ g is isomorphic to e, with the isomorphism deﬁned by r Θ ∗ ϕ → ϕ(e). Now we show that all the ϕ just constructed are regular differential forms, that is, are elements of Ω r [G]. In view of their invariance, it is enough to show that the forms are regular at any point, for example at the identity element e. Moreover, it is enough to restrict ourselves to the case r = 1. Indeed, if η = ∧ · · · ∧ αir, with αj ∈ e, and we prove that the forms ϕj corresponding by (3.69) to the αj are regular, then the form ϕ = ∧ · · · ∧ ϕir is regular, and it corresponds by (3.69) to η. 1 Θ ∗ αi1 ϕi1 We choose an afﬁne neighbourhood V of e such that Ω 1[V ] is free over k[V ], and write du1,..., dun for a basis. Then there exists an afﬁne neighbourhood U of e such that μ(U × U ) ⊂ V, where μ : G × G → G is the multiplication map of G. Just as any function of k[U × U ], the elements μ∗(ul) can be written in the form μ∗(ul)(g1, g2) = vlj (g1)wlj (
g2) for (g1, g2) ∈ U × U ⊂ G × G, 204 3 Divisors and Differential Forms where vlj, wlj ∈ k[U ]. By deﬁnition, th = μ ◦ sh, where sh is the embedding G → G × G given by sh(g) = (h, g). Hence t ∗ h (ul)(g) = vlj (h)wlj (g), and since t ∗ t ∗ t ∗ (g) = (g) = dhg vlj (h)dhg(wlj ). In particuh (dul) h (ul) h (dul) lar, setting h = g−1, we get, we have t ∗ g−1(dul) (g) = g−1 dewlj. vlj Expressing dwlj in terms of duk, we obtain the relations t ∗ g−1(dul) = k ckl(g)duk, with ckl ∈ k[U ], (3.70) where ckl(g) = j g−1 vlj ∂wlj ∂uk (e). (3.71) Now write the invariant form ϕ in the form u = ψkduk and consider the rela- tion t ∗ g (ϕ) = ϕ at e. Substituting (3.70) and equating coefﬁcients of duk, we get cklψl = ψk(e). (3.72) Since (ckl(e)) is the identity matrix, we get det \|ckl\|(e) = 0, and it follows from the system of equations (3.72) that ψk ∈ Oe. We state the result we have proved: Proposition The map ω → ω(e) establishes an isomorphism from the vector space of invariant regular differential r-forms on G to r Θ ∗ e. 6.3 The Canonical Class We now consider more particularly rational differential n-forms on an n-dimensional nonsingular variety X (compare Section 5.3). In some neighbourhood of a point x ∈ X, such a form can be written ω = gdu1 ∧ · · · ∧ dun. We
cover X by afﬁne sets Ui such that on each Ui we have such an expression ω = g(i)du(i) ∧ · · · ∧ du(i) n. 1 On the intersection Ui ∩ Uj, by (3.63), we get g(j ) = g(i)J u(i) 1,..., u(i) 1,..., u(j ) u(j ) n n. Since the Jacobian determinant J is regular and nowhere zero in Ui ∩ Uj (see (3.62)), the system of functions g(i) on Ui is a compatible system of functions in the sense of Section 2.1, and hence deﬁnes a divisor on X. This divisor is called the divisor of ω, and is denoted by div ω. 6 Examples and Applications of Differential Forms 205 The following properties of the divisor of a rational differential n-form on a non- singular n-dimensional variety follow at once from the deﬁnition: (a) div(f ω) = div f + div ω for f ∈ k(X). (b) div ω ≥ 0 if and only if ω ∈ Ω n[X]. By the case r = n of Theorem 3.19, Ω n(X) is a 1-dimensional vector space over k(X). Hence if ω1 ∈ Ω n(X) and ω1 = 0, then any form ω ∈ Ω n(X) can be written ω = f ω1. Hence property (a) shows that the divisors of all forms ω ∈ Ω n(X) are linearly equivalent, and form one divisor class on X. This class is called the canonical class of X, and is denoted by K or KX. Let ω1 ∈ Ω n(X) be a ﬁxed n-form, so that any other can be written f ω1. (b) shows that ω is regular on X if and only if div f + div(ω1) ≥ 0. In other words, in terms of the notion of vector space associated with a divisor introduced in Section 1.5, we have Ω n
[X] = L(div(ω1)). Thus hn = dimk Ω n[X] = (KX). We see that the invariant hn introduced in Section 6.1 is equal to the dimension of the canonical class. r Θ ∗ Example Suppose that X is the variety of an algebraic group. We showed in Section 6.2 that the vector space of invariant differential r-forms on X is isomorphic to e, where Θe is the tangent space to X at the identity element e. In particular, ∼= k. If the space of invariant differential n-forms is 1-dimensional, since ω is a nonzero invariant form then ω ∈ Ω n[X], that is div ω ≥ 0. But if ω(x) = 0 for some point x ∈ X, then by invariance also ω(y) = 0 for every y ∈ X. Hence ω(x) = 0 for all x ∈ X, that is, ω is regular and nowhere vanishing on X. This means that div ω = 0, that is KX = 0. n Θ ∗ e In Theorem 3.9, we proved that the number (D) is ﬁnite for any divisor D on a nonsingular projective algebraic curve. It follows in particular that the number h1 = dimk Ω 1[X] = (KX) is ﬁnite for any nonsingular projective algebraic curve X. This number is called the genus of the curve X, and denoted by g = g(X); that is, for curves, h1 = g. There are several other characterisations of the genus of a curve, see for example Corollary 3.1 and Section 3.3, Chapter 7. In the case dim X = 1 we know that all divisors in one linear equivalence class have the same degree, so that it makes sense to speak of the degree deg C of a divisor class C. In particular, the degree deg KX of the canonical class is a birational invariant of a curve X. The invariants g(X) and deg KX we have introduced are not independent. It can be proved that the relation deg KX = 2g(X) − 2 holds between them; see Corollary 3.1. In
particular, if a nonsingular projective curve X is an algebraic group then KX = 0, as we have just seen. Hence g(X) = 1, that is, of all projective curves, only the curves of genus 1 can have an algebraic group law deﬁned on them. We will see in Corollary 3.4 that the curves of genus 1 are exactly the nonsingular cubic curves. 206 3 Divisors and Differential Forms 6.4 Hypersurfaces We now compute the canonical class and the invariant hn(X) = (KX) in the case that X ⊂ PN is a nonsingular n-dimensional hypersurface, with N = n + 1. Suppose that X is deﬁned by the equation F (x0 : · · · : xN ) = 0 with deg F = deg X = m. Consider the afﬁne open set U with x0 = 0. Our X is deﬁned in U by G(y1,..., yN ) = 0, where yi = xi/x0 and G(y1,..., yN ) = F (1, y1,..., yN ). Deﬁne the open subset Ui ⊂ U by ∂G/∂yi = 0; then y1,...,yi,..., yN (with yi omitted) are local parameters in Ui, and the form dy1 ∧ · · · ∧ dyi ∧ · · · ∧ dyN is a basis of Ω n[Ui] over k[Ui]. However, it is more convenient to take as basis the form ωi = (−1)i ∂G/∂yi dy1 ∧ · · · ∧ dyi ∧ · · · ∧ dyN, which is permissible, since ∂G/∂yi = 0 in Ui. The advantage is that then the forms ω1,..., ωN are equal: multiplying the relation by the product dy1 ∧ · · · ∧ dyi dyi ∧ dyi = 0 we see that N ∂G ∂yi dyi = 0 i=1 ∧ · · · ∧ dyj ∧ · · · ∧ dyN
, and using the fact that ωj = ωi. (3.73) Since X is nonsingular, U = Ui, and it follows from (3.73) that the ωi ﬁt together to give a form ω that is regular and everywhere nonzero on U, so that div ω = 0 in U. It remains to study points not in U. Consider, say, the open subset V in which x1 = 0. This afﬁne space has coordinates z1,..., zN with z1 = 1 y1 y1 = 1 z1 and zi = yi y1 and yi = zi z1 for i = 2,..., N; for i = 2,..., N. Hence dy1 = − dz1 z2 1 and dyi = z1dzi − zidz1 z2 1 for i = 2,..., N. We substitute these expressions in ωN, and use the fact that dz1 ∧ dz1 = 0, obtaining ω = − (−1)N zN 1 (∂G/∂yN ) dz1 ∧ · · · ∧ dzN −1. 6 Examples and Applications of Differential Forms 207 The equation of X in V is H (z1,..., zN ) = 0, where H = zm 1 G 1 z1, z2 z1,...,. zN z1 From the relation ∂H ∂zN = zm−1 1 ∂G ∂yN it follows that 1 z1, z2 z1,..., zN z1 = zm−1 1 ∂G ∂yN (y1,..., yN ) ω = − (−1)N zN −m+1 1 (∂H /∂zN ) dz1 ∧ · · · ∧ dzN −1. (3.74) All the arguments that we carried out for U are also valid for V, and show that Ω n[V ] = k[V ] 1 ∂H /∂zN dz1 ∧ · · · ∧ dzN −1. (3.75) Hence in V we have div ω = −(N −
m + 1) div(z1). Obviously div(z1) in V is the divisor of the form x0 on X, as deﬁned in Section 1.2. Finally, we get that the relation div ω = (m − N − 1) div(x0) = (m − n − 2) div(x0). Thus KX is the divisor class containing (m − n − 2)L, where L is the hyperplane section of X. We now determine Ω n[X]. Writing any form η ∈ Ω n(X) as η = ϕω, we see that η ∈ Ω n[X] if and only if ϕ ∈ L((m − n − 2) div(x0)). By Example of Section 1.5, this is equivalent to ϕ = P (z1,..., zn), where P is a polynomial, and deg P ≤ m − n − 2. (3.76) From this it is easy to compute the dimension of Ω n[X]. Namely two different polynomials P, Q ∈ k[y1,..., yN ] satisfying (3.76) deﬁne different elements of k[X], since otherwise P − Q is divisible by Q, which contradicts (3.76). Thus the dimension of Ω n[X] equals the dimension of the space of polynomials P satisfying = (m−1)···(m−N ) (3.76). This dimension is equal to the binomial coefﬁcient. N! Thus m−1 N hn(X) = (KX) =. (3.77) m − 1 n + 1 The simplest case of this formula is when N = 2, that is, n = 1. We get the formula g(X) = m − 1 2 = (m − 1)(m − 2) 2 for the genus of a nonsingular plane curve of degree m. Compare Example 4.10. We can make an important deduction at once from (3.77). Namely, interpreting the binomial coefﬁcients as the number of combinations, we get at once that m − 1 n + 1 for m > m ≥ n + 1. m − 1 n + 1 > 208 3 Divis
ors and Differential Forms Since by Theorem 3.22, hn(X) is a birational invariant, (3.77) thus implies that hypersurfaces of different degrees m, m ≥ n + 1 are not birational. We see that there exist inﬁnitely many algebraic varieties of any given dimension not birational to one another. In particular, when N = 2, m = 3 we get g(X) = 1, and since g(P1) = 0, we see once again that a nonsingular cubic curve in P2 is not rational. It follows from (3.77) that hn(X) = 0 if m ≤ N. In particular, hn(Pn) = 0. We veriﬁed this directly for n = 1 in Example 3.15. Consider the case m ≤ N in more detail. If N = 2 then this means m = 1 or 2. For m = 1 we have X = P1, and we already know that h1(P1) = 0. For m = 2 we have a nonsingular plane curve of degree 2, which is isomorphic to P1, so that in this case also h1(X) = 0 tells us nothing new. Suppose that N = 3. For m = 1 we have P2, and we already know that h2 = 0. If m = 2 then X is a nonsingular surface of degree 2, which is birational to P2, so that h2(X) = 0 is a consequence of h2(P2) = 0 and Theorem 3.22. If m = 3 then X is a nonsingular cubic surface. If such a surface contains two skew lines then it is birational to P2 (Example 1.23). One can show that every nonsingular cubic surface contains two skew lines, so that h2(X) = 0 is again a consequence of h2(P2) = 0 and Theorem 3.22. The examples we have considered lead to interesting questions on nonsingular hypersurfaces of small degree, X ⊂ PN with deg X = m ≤ N. We see that for N = 2, 3, X is birational to projective space Pn, with n = N − 1, which “explains” the equality hn(X) = 0. For N = 4 we run into a
new phenomenon. For m = 3, for example, already for the cubic hypersurface X given by x3 0 + x3 1 + x3 2 + x3 3 + x3 4 = 0, (3.78) the question of whether X is birational to P3 is very subtle. However, one can show that there exists a rational map ϕ : P3 → X such that ϕ(P3) is dense in X and k(X) ⊂ k(P3) is a separable extension (see Exercise 20). Already this, together with h3(P3) and Theorems 3.21–3.22, implies h3(X) = 0. The following terminology arises in connection with this: we say that a variety is rational if it birational to Pn where n = dim X, and unirational if there exists a rational map ϕ : Pn → X such that ϕ(Pn) is dense in X and k(X) ⊂ k(Pn) is separable. It follows from Exercise 6 and Theorems 3.21–3.22 that all hi = 0 for a unirational variety. The question of whether the notions of rational and unirational varieties are the same is typical of a series of problems arising in algebraic geometry. This question is called the Lüroth problem. It can obviously be stated as a problem in the theory of ﬁelds: if K is a subﬁeld of the rational function ﬁeld k(T1,..., Tn) such that K ⊂ k(T1,..., Tn) is a ﬁnite separable extension, then is it true that K is isomorphic to the rational function ﬁeld? For n = 1 the answer is positive, in fact even without the assumptions that k is algebraically closed and K ⊂ k(T ) is separable. For n = 2 the answer is negative without these assumptions, but positive with them, but the proof is very delicate. 6 Examples and Applications of Differential Forms 209 It is given, for example, for ﬁelds of characteristic 0 in Shafarevich [69, Chapter III] or Barth, Peters and van de Ven [9], and in the general case in Bombieri and Husemoller [14]. For
n ≥ 3 the answer is negative even if k = C. This is a delicate result of the theory of 3-folds. One of the examples of a unirational but not rational variety is the nonsingular cubic 3-fold, in particular the hypersurface (3.78) (see Clemens and Grifﬁths [23]). Another example of an irrational variety is the nonsingular quartic hypersurface of P4 (see Iskovskikh and Manin [43]); some of these hypersurfaces are unirational. For another type of example, see Artin and Mumford [7]. Whereas for 3-folds the Lüroth problem is a subtle geometric problem, in higher dimension it turns out to be more algebraic in spirit, and its solution more elementary. For example, there are examples of ﬁnite group G of linear transformations of variable x1,..., xn such that the subﬁeld of invariants k(x1,..., xn)G of this group is not isomorphic to the ﬁeld of rational functions (see Bogomolov [13] or Saltman [67]). 6.5 Hyperelliptic Curves As a second example we consider one type of curves. Write Y for the afﬁne plane curve with equation y2 = F (x), where F (x) is a polynomial with no multiple roots and of odd degree n = 2m + 1 (we proved in Section 1.4, Chapter 1 that the case of even degree reduces to the odd degree case). We suppose that char k = 2. The nonsingular projective model X of Y is called a hyperelliptic curve. We compute the canonical class and the genus of X. The rational map Y → A1 given by (x, y) → x deﬁnes a regular map f : X → P1. Obviously deg f = 2, so that, by Theorem 3.5, if α ∈ P1 and u is a local parameter at α, the inverse image f −1(α) either consists of two points z, z with vz(u) = vz(u) = 1, or f −1(α) = z with vz(u) = 2. It is easy to check that the afﬁne curve Y is nonsingular
. If Y is its projective closure in P2 then X is the normalisation of Y, and we have a map ϕ : X → Y which is an isomorphism of ϕ−1(Y ) and Y. It follows that if ξ ∈ A1 has coordinate α then * f −1(ξ ) = {z, z} z if F (α) = 0; if F (α) = 0. Now consider the point at inﬁnity α∞ ∈ P1. If x denotes the coordinate on P1 then a local parameter at α∞ is u = x−1. If f −1(α∞) = {z, z} consisted of 2 points then u would be a local parameter at z, say. It would follow that vz(u) = 1 and hence vz(F (x)) = −n; but since y2 = F (x), we have vz(F (x)) = 2vz(y), and this contradicts n odd. Thus f −1(α∞) consists of just one point z∞, and vz∞(x) = −2, vz∞(y) = −n. It follows that X = ϕ−1(Y ) ∪ z∞. We proceed to differential forms on X. Consider, for example, the form ω = dx/y. At a point ξ ∈ Y, if y(ξ ) = 0 then x is a local parameter, and vξ (ω) = 0. 210 3 Divisors and Differential Forms If y(ξ ) = 0 then y is a local parameter, and vξ (x) = 2, so that it again follows that vξ (ω) = vξ (dx)−vξ (y) = 1−1 = 0. Thus div ω = kz∞, and it remains to determine the value of k. For this, it is enough to recall that if t is a local parameter at z∞ then x = t −2u and y = t −nv, where u, v, u−1, v−1 ∈ Oz∞. Hence ω = t n−3wdt with w, w−1 ∈ Oz∞, and therefore div ω = (n − 3)z∞
= (2m − 2)z∞. Now we determine Ω 1[X]. As we have seen, ω is a basis of the module Ω 1[Y ], that is, Ω 1[Y ] = k[Y ]ω, so that any form in Ω 1[X] is of the form uω, where u ∈ k[Y ], and hence u is of the form P (x) + Q(x)y with P, Q ∈ k[X]. It remains to see when these forms are regular at z∞. This happens if and only if vz∞(u) ≥ −(n − 3). (3.79) We ﬁnd all such u ∈ k[Y ]. Since vz∞(x) = −2, it follows that vz∞(P (x)) is always even and since vz∞(y) = −n, that vz∞(Q(x)y) is always odd. Hence vz∞(u) = vz∞ P (x) + Q(x)y ≤ min vz∞ P (x), vz∞ Q(x)y and so if Q = 0 we have vz∞(u) ≤ −n. Thus u = P (x) and (3.79) gives 2 deg P ≤ n − 3, that is, deg P ≤ m − 1, where n = 2m + 1. We have found that Ω 1[X] consists of forms P (x)dx/y where the degree of P (x) is ≤m − 1. It follows from this that g(X) = h1(X) = dim Ω 1[X] = m. It is interesting to compare the results of Section 6.4 when N = 2 and of Section 6.5. In the second case we saw that there exist algebraic curves of any given, that is, it is genus. In the ﬁrst case, the genus of a nonsingular plane curve is a long way from giving an arbitrary integer. Thus not every nonsingular projective curve is isomorphic to a plane curve. For example, a hyperelliptic curve of genus 4, with n = 9 is not. n−1 2 7 The Riemann–Roch Theorem on Curves 7.1 Statement
of the Theorem In this section we prove and discuss one of the central results of the theory of algebraic curves. This is the Riemann–Roch theorem, that consists of the following statement. Theorem 3.23 For an arbitrary divisor D on a nonsingular projective algebraic curve X, we have the relation (D) − (K − D) = deg D − g + 1, (3.80) where K is the canonical divisor of X and g its genus. 7 The Riemann–Roch Theorem on Curves 211 The relation (3.80) is not directly a formula for the number (D) that we are interested in, but rather for the difference (D) − (K − D). It turns out that this is a “rough” invariant of D, depending only on the degree deg D, whereas (D) itself is a “ﬁner” invariant, that may take different values for different divisors D of the same degree. Thus on a nonsingular cubic curve X, if deg D = 0 then (D) may take different values: if D = p − q with p, q distinct points then (D) = 0, but if D = 0 then (D) = 1. The higher dimensional generalisation of the relation (3.80) is similar in character: its left-hand side is a sum with alternating signs of certain invariants of D; for an n-dimensional variety the expression has n + 1 terms. It is only this combination of the terms that is the “rough” invariant of D (in a sense that needs to be deﬁned more precisely—it “does not change under small perturbations” of D). In a different direction, the generalised form of (3.80) relates to the theory of elliptic operators on manifolds. Here again we are talking about a difference of two quantities, the index of an elliptic operator—the difference in dimension of its kernel and cokernel. There are results analogous to the Riemann–Roch theorem in other questions, for example, in topology. In view of this, we devote a separate section to it. However, we must ﬁrst explain the interest of the result for curves, since the relation (3.80) expresses the number (D) we seek in terms of (K −
D), which at ﬁrst sight seems to be no improvement on (D). To answer this point, we give a series of applications of the theorem. Corollary 3.1 If we set D = K then, using (K − K) = (0) = 1 and (K) = g, we get that deg K = 2g − 2. We discussed this equality in Section 6.3. Corollary 3.2 If deg D > 2g − 2 then (D) = deg D − g + 1. This follows because deg D > 2g − 2 gives deg(K − D) < 0, which implies that (K − D) = 0: indeed, an effective divisor K − D ∼ D ≥ 0 would contradict deg D < 0. Corollary 3.3 If g = 0 then X ∼= P1. Indeed, let D = x ∈ X be a point; then (3.80) gives that (D) ≥ 2. This implies that the space L(D) contains a nonconstant function f in addition to the scalars. Any such function has div(f )∞ = x; that is, if we interpret f as a map f : X → P1 then deg f = 1 by Theorem 3.5. It follows that X ∼= P1. In other words, g = 0 is not only a necessary condition, but also a sufﬁcient condition for X to be a rational curve. Corollary 3.4 If g = 1 then X is isomorphic to a cubic curve in P2. Indeed, if g = 1 then Corollary 3.2 gives (D) = deg D for D > 0, and the as- sertion follows from Theorem 3.11. 212 3 Divisors and Differential Forms Corollary 3.5 For D ≥ 0, consider a basis f0,..., fn of L(D) and the corresponding map ϕ = (f0,..., fn) : X → Pn. We determine when ϕ is an embedding. We prove that this holds if and only if the conditions (D − x) = (D) − 1 and (D − x − y) = (D) − 2 (3.81) hold for all points x, y ∈ X. It follows from Corollary 3.2 that the equal
ities (3.81) hold if deg D ≥ 2g + 1, so that ϕ is then an embedding. Proof Note that the ﬁrst condition in (3.81) guarantees that −D = GCD(div fi) is the greatest common divisor of the div fi. Indeed, div fi ≥ −D by deﬁnition; however, if GCD(div fi) > −D then there would exist a point x such that GCD(div fi) > −D + x, so that L(D) ⊂ L(D − x), hence (D) ≤ (D − x), which contradicts (3.81). Therefore, by the remark at the end of Section 1.4, the divisors are the inverse images of the hyperplanes of Pn under the map ϕ. Dλ = div To prove that ϕ is an isomorphic embedding, we use Lemma of Section 5.4, Chapter 2 and Theorem 2.24, whose assumptions we can check using the above remark. If ϕ(x) = ϕ(y) then every hyperplane E that passes through ϕ(x) also passes through ϕ(y). This means that if Dλ − x ≥ 0 then also Dλ − x − y ≥ 0, that is, (Dλ − x) ≤ (Dλ − x − y), which contradicts (3.81). i λifi We prove that the tangent spaces at a point is mapped isomorphically. This is equivalent to saying that ϕ∗ : mϕ(x)/m2 ϕ(x) → mx/m2 x is surjective. If this does not hold then ϕ∗(mϕ(x)) ⊂ m2 x, because in our case = 1. In other words, any u ∈ mϕ(x) satisﬁes vx(ϕ∗(u)) ≥ 2. If we apdim mx/m2 x ply this to linear functions, this shows that if Dλ − x ≥ 0 then also Dλ − 2x ≥ 0. We again deduce that (D − x) ≤ (D − 2x), which contradicts the second condition in (3.81). This completes the proof of Corollary 3.5. Obviously, changing the choice of basis of L(D
) changes ϕ by composing it with a projective linear transformation of Pn. On the other hand, replacing D by a linearly equivalent divisor D + div f corresponds to mapping L(D) by the isomorphism u → uf, and hence does not change ϕ. Thus it makes sense to speak of the map ϕ corresponding to a divisor class. Suppose, for example, that X is a curve of genus 1, and x0 ∈ X. By Corollary 3.2, 3x0 satisﬁes the conditions (3.81) of Corollary 3.5. Hence the map ϕ corresponding to 3x0 is an isomorphism of X to a curve Y ⊂ P2 (since (3x0) = 3 by Corollary 3.2). As we have seen, 3x0 is the pullback of a section of Y by a line, and since deg 3x0 = 3, also deg Y = 3. Thus every curve of genus 1 is isomorphic to a plane cubic. (For more details, compare the proof of the converse part of Theorem 3.11.) The most interesting maps ϕ are those corresponding to classes intrinsically related to X, for example, the multiples nK of the canonical class. Corollary 3.1 shows that deg nK ≥ 2g + 1 for n ≥ 2 if g > 2, and for n ≥ 3 if g = 2. Thus for g > 1 the class 3K always satisﬁes (3.81) of Corollary 3.5. The corresponding map ϕ3K take 7 The Riemann–Roch Theorem on Curves 213 X to Pm where m = (3K) − 1 = 5g − 6 (by Corollary 3.2). Under this embedding, two curves X and X are isomorphic if and only if their images ϕ3K (X) and ϕ3K (X) are obtained from one another by a projective linear transformation of Pm. This reduces the question of the birational classiﬁcation of curves to a projective classiﬁcation. The map ϕK corresponding to the canonical class itself is not always an embed- ding; however, all the cases when this fails are enumerated (see Exercises 18–19). As a simple application of these ideas,
consider a nonsingular plane curve of degree 4. By Section 6.4, its canonical class is the divisor class of the intersection of X with a line of P2. Hence the map ϕK corresponding to the canonical class is just the natural embedding in the plane. It follows from what we have said that two such curves are isomorphic if and only if they are projectively equivalent. This leads us to an extremely important conclusion. We can identify the set of plane 6 − 1, as in Example 1.28. One sees that curves of degree 4 with P14, where 14 = 2 the nonsingular curves form an open subset of the same dimension. On the other hand, the group of all projective transformations of the plane (that is, nondegenerate 3 × 3 matrices up to constant multiples) has dimension 8. From this one deduces that P14 has an open subset U and a map f : U → M to a certain variety M such that two points u1, u2 ∈ U parametrise projectively equivalent curves if and only if they belong to the same ﬁbre of f. The dimension of the ﬁbre is 8, so that dim M = 14 − 8 = 6. Thus two plane curves of degree 4 are by no means always isomorphic: to be isomorphic, they must correspond to the same point of a 6-dimensional variety M. This shows that the genus is not a complete system of birational invariants of curves. In addition to the integer invariant, the genus, curves also have “continuous” invariants called moduli. It can be proved that the set of all curves of given genus g > 1 form (in a sense that we do not make precise) a single irreducible variety of dimension 3g − 3. In the case of plane quartics, the genus g = 3 and 3g − 3 = 6 = dim M. A similar thing happens for curves of genus 1 (see Exercise 8 of Section 3.6). It is only for g = 0 that all curves of the same genus are isomorphic. 7.2 Preliminary Form of the Riemann–Roch Theorem We now prove a certain auxiliary relation, outwardly similar to (3.80), that will later allow us to deduce the Riemann–Roch theorem itself. Deﬁnition A distribution on an irreducible nonsingular
curve X is a function that assigns to every point x ∈ X a certain rational function rx ∈ k(X), with the condition that vx(rx) ≥ 0 for all except possibly ﬁnitely many points of X. The sum of distributions and their multiplication by elements of k is deﬁned by these operations applied to the rx. For example, if two distributions r and s assign 214 3 Divisors and Differential Forms the functions rx and sx to x then (r + s)x = rx + sx. One sees easily that under these operations, distributions form a vector space over k (highly inﬁnite dimensional), that we denote by R. By analogy with the space L(D) associated with the divisor D introduced in Section 1.5, we deﬁne the set R(D) as the set of all distributions r for which nixi, and vx(rx) ≥ 0 if x = xi. Obviously R(D) ⊂ vxi (rxi ) + ni ≥ 0, where D = R is a vector subspace. We assign to each function f ∈ k(X) the distribution r with rx = f for every x ∈ X. It is clear that this correspondence deﬁnes an embedding of k(X) into R as a k-vector subspace. We use the same letter f to denote the function f and the corresponding distribution. Starting from the two subspaces R(D) and k(X), we can form the vector subspace R(D) + k(X) of R, consisting of sums of elements of R(D) and k(X). Theorem 3.24 For any divisor D on a nonsingular projective curve X, the space Λ(D) = R/(R(D) + k(X)) is ﬁnite dimensional. For D ≥ D, consider the quotient space R(D) + k(X) R(D) + k(X). / We start by proving that it is ﬁnite dimensional. Moreover, we prove the following relation that will be important in what follows. Lemma If D ≥ D are divisors on a nonsingular projective curve, the quotient (R(D) + k(X))/(R(D) + k(X
)) is ﬁnite dimensional, and its dimension is given by R dim D R(D) + k(X) + k(X) / = D deg D − − deg D − (D). (3.82) To prove this, we ﬁrst consider the space R(D)/R(D), and prove that it is ﬁnite dimensional, or more precisely, that R D /R(D) dim = deg D − deg D. The proof is practically the same as that of the Lemma in Section 2.3. Suppose that D − D = miPi. Then for a distribution r ∈ R(D), the condition r ∈ R(D) can be written as a separate condition for each pi. If the Laurent expansion rpi at pi in a local parameter t is amt m + · · · + am+mi −1t m+mi −1 + · · ·, the condition is that am = · · · = am+mi −1 = 0. Here m = ν(rpi ). The number of these conditions at each point pi equals mi, so that there are altogether 215 7 The Riemann–Roch Theorem on Curves mi = deg D − deg D of them. The linear independence of the conditions is ob- vious (compare Exercise 4 of Section 2.4, Chapter 2). We now proceed to the proof of (3.82). Consider the standard homomorphism D R → R D R(D) + k(X) + k(X), / which is surjective. Its kernel obviously equals R(D) ∩ (R(D) + k(X)). But if A, B, C are any three subspaces of a vector space with A ⊃ B (or even just subgroups of an Abelian group), one checks at once that A ∩ (B + C) = B + (A ∩ C). On the other hand, the equality R(D) ∩ k(X) = L(D) is a tautology. As a consequence, we see that R R(D) + k(X) / + k(X) R(D) + L D D D / ∼= R. Consider the vector space R(D)/R(D), whose dimension we know. It contains (R(D) + L(D))/R
(D), the surjective image of L(D) with kernel L(D). Therefore (R(D) + L(D))/R(D) ∼= L(D)/L(D). We deduce from this that the relation (3.82) in the Lemma holds. In preparation for the proof of Theorem 3.23, we need to establish that deg D − (D) is bounded above for all divisors D on a curve X, which follows in a straightforward way from the material of Section 2.1. To see this, choose any nonconstant rational function f ∈ k(X), and use it to deﬁne a regular map f : X → P1. By deﬁnition, its degree is n = deg f = k(X) : k P1. Arguing as in Theorems 3.6 and 3.7, let α1,..., αn be a basis of k(X) over k(P1) = k(f ), and write Dα for the least common divisor of poles of the αi, so that div(αi) + Dα ≥ 0 for each i. Also, write A for the divisor of poles of f, so that A = f ∗(∞), and deg A = n by Theorem 3.5. Lemma (I) We have αi · f j ∈ L(Dα + mA) for i = 1,..., n and j ≤ m, (3.83) and these elements are linearly independent. In particular (Dα + mA) ≥ (m + 1)n, whereas deg(Dα + mA) = deg Dα + mn, so that deg(Dα + mA) − (Dα + mA) is bounded above by deg(Dα + mA) − (Dα + mA) ≤ deg D0 − n. (3.84) (II) Every divisor D on X is dominated by a divisor linearly equivalent to mA for some integer m. 216 3 Divisors and Differential Forms Proof (I) Each αi · f j with j ≥ m has divisor of poles at most Dα + mA, by deﬁnition of Dα and A. The elements (3.83
) are linearly independent by construction: f ∈ k(X) is transcendental over k, and α1,..., αn ∈ k(X) are linearly independent over k(f ). (II) is an easy veriﬁcation. The lemma is proved. Corollary On a ﬁxed curve X, the difference deg(D) + 1 − (D) is bounded above for all D. Proof This follows by Remark 3.3. Every divisor D is dominated by a divisor linearly equivalent to Dα + mA, to which (3.84) applies, so that the statement follows from (3.26). We can now proceed with the proof of Theorem 3.24. We know that (D)−deg D is bounded below for all divisors D on a curve X. Consider some divisor D0 for which this difference takes the smallest possible value. Then the inequality (D) − deg D ≥ (D0) − deg D0 holds for any divisor D. In particular, setting D = D0 and D ≥ D in (3.82), we get that (D) − deg D = (D0) − deg D0, or in other words, R(D) + k(X) = R(D0)+k(X). But it is obvious that for any distribution r ∈ R, there exists a divisor D such that r ∈ R(D), and equally obvious that we can choose D ≥ D0. Therefore R(D) = R, where the union runs over all D ≥ D0, and a fortiori, we have (R(D) + k(X)) = R. It follows from this that R(D0) + k(X) = R, and hence Λ(D0) = 0. For any divisor D it is easy to ﬁnd a divisor D1 such that D1 ≤ D and D1 ≤ D0. From this, applying (3.82) ﬁrst to D = D0 and D = D1, we get the ﬁnite dimensionality of λ(D1), and then applying it to D = D and D = D1, that of Λ(D). This justiﬁes us in writing dim Λ(D
) = λ(D). The preliminary form of the Riemann–Roch theorem is the following. Theorem 3.25 For any divisor D on a smooth curve X, the following equality holds: (D) − λ(D) = deg D − λ(0) + 1. This is practically a restatement of the Lemma after Theorem 3.24: ﬁrst, since we know that Λ(D) is ﬁnite dimensional, we can restate the Lemma as saying that for D ≥ D, λ(D) − λ D = D deg D − − deg D − (D), or D D − λ = (D) − λ(D) + deg D − deg D. (3.85) 7 The Riemann–Roch Theorem on Curves 217 We have already proved this under the condition that D ≥ D, but it follows from this that it holds for any divisors D and D. For this, we need only take a divisor D such that D ≥ D and D ≥ D, write out the relations (3.85) for both pairs of divisors D, D and D, D, and subtract one from the other. Finally, since (3.85) is proved for any divisors D (and any D), we can substitute D = 0 in it. Since (0) = 1 and deg 0 = 0, we obtain the equality of Theorem 3.25. 7.3 The Residue of a 1-Form The Riemann–Roch theorem will follow from Theorem 3.25 if we can prove that λ(D) = (K − D), since λ(0) = g. Both numbers are deﬁned as the dimension of certain vector spaces Λ(D) and L(K − D), so it is natural to propose that the equality λ(D) = (K − D) is a reﬂection of some assertion on the vector space themselves. Indeed, we will prove that Λ(D) and L(K − D) are dual vector spaces. For this, we need to construct a pairing between them, that is, a function (u, v) of u ∈ Λ(D) and v ∈ L(K − D) taking values in k, and satisfying the conditions
of linearity and nondegeneracy in both arguments; that is, (u1 + u2, v) = (u1, v) + (u2, v), and (u, v) = 0 for all v ∈ L(K − D) (αu, v) = α(u, v) for α ∈ k, if and only if u = 0, (3.86) and the analogous properties in the second argument. This section contains some introductory ideas that clarify the deﬁnition of the pairing (u, v). We defer the deﬁnition itself to later. We consider rational 1-forms on a curve X, that is, in the notation of Section 5.4, elements of the space Ω 1(X). Using the notation of the divisor div ω of a form ω ∈ Ω 1(X) from Section 6.3, one checks easily that the space L(K − D) is isomorphic to the space of forms ω for which div ω ≥ D. We denote this space by Ω 1(D). Thus in what follows we are talking about a pairing between distributions and certain differential forms. This relates to a new notion. Choosing a local parameter t at a point x ∈ X, we can write a 1-form ω as an expression f dt with f ∈ k(X). We consider the formal Laurent series expansion of f at x: f = a−mt −m + · · · + a0 + a1t + · · · + ant n + · · ·. (3.87) Deﬁnition The coefﬁcient a−1 in this expansion is called the residue of the 1-form ω at x, and written Resx ω. The unexpected thing about this notion is that it is intrinsic; that is, Resx ω depends only on the point x and the 1-form ω, and not on the choice of the local parameter t at x. Over a ﬁeld of characteristic 0, the residue is obviously well deﬁned up to a scalar α ∈ k: indeed, one checks easily that the differential map 218 3 Divisors and Differential Forms f → df extends from k(X) to the formal Laurent series ﬁeld k((t)), and the image of d :
k((t)) → k((t)) consists of the expressions ω = f dt with Res ω = 0 (that is, the image contains every t kdt with k = −1). Thus Res ω is a linear form on k((t))dt having a well deﬁned kernel, so that the form is well deﬁned up to multiplication by an element of k. Its complete invariance requires a small computation even in the case of characteristic 0. Namely, let u be another local parameter at x (that is, an element of the formal power series ring k[[t]] with an expansion of the form u = α1t + · · · + αnt n + · · · with α1 = 0). In other words, u ∈ Ox, u(0) = 0 and ∂u = 0. Then conversely, t ∈ k[[u]] has an expansion t = β1u + · · · + βnun + · · · with ∂t β1 = 0. We must substitute this expression for t in f dt given by (3.87), and ﬁnd the coefﬁcient of u−1du. We already know that the terms akt kdt with k = −1 do not contribute anything, so we only need to consider the single term a−1t −1dt. In this −1 × u−1 × v where v ∈ k[[u]] has v(0) = 0, and dt = (β1 + 2β2 +... )du. t −1 = β 1 As a result of the substitution, the coefﬁcient of u−1du remains unchanged, with the −1 β1 and β 1 cancelling out. This argument (or at least, the ﬁrst part of the argument) does not work for a ﬁeld of ﬁnite characteristic. There are other, much more complicated, proofs using speciﬁc properties of the case of ﬁnite characteristic. It is only when the ground ﬁeld is the complex number ﬁeld C, when the curve is 2-dimensional as a real surface, that we have a completely intrinsic deﬁnition: Resx ω = 1 C ω, where C is a 2πi sufﬁ
ciently small contour going once around x. (More precisely, it follows in this case from the invariance of the integral.) We give below a deﬁnition of the residue that is valid over a ﬁeld of arbitrary characteristic. + We observe that we are now in a position to deﬁne a pairing between the space R of distributions and the space Ω 1(X) of differential 1-forms, by setting (r, ω) = x Resx(rxω), (3.88) where the sum runs over all points x ∈ X. This sum makes sense, since for all but a ﬁnite number of points x we have rx ∈ Ox, and ω and the form rxω are regular at x, so that there are no negative terms in (3.87). Only points at which these conditions fail can contribute to the sum in (3.88). In view of Λ(D) = R/(R(D) + k(X)), this construction deﬁnes a function (u, v) for u ∈ Λ(D) and v ∈ Ω 1(D) provided that the following two conditions hold: (1) (R(D), Ω 1(D)) = 0, (2) (k(X), Ω 1(D)) = 0. The ﬁrst condition is obvious: for r ∈ R(D) and ω ∈ Ω 1(D) the form rxω is regular at x, so that Resx(rxω) = 0 for all x. While much less obvious, the second condition is nonetheless true. Since the ﬁrst argument does not depend on D, the condition has the form (k(X), Ω 1(X)) = 0. The distribution corresponding to f ∈ k(X) has all rx equal to f, and the required x Resx(f ω) = 0. But in our case, f ω is any 1-form of relation takes the form 7 The Riemann–Roch Theorem on Curves 219 Ω 1(X), and thus we arrive at the relation x Resx ω = 0 for all ω ∈ Ω 1(X). (3.89) This equality is known as the residue theorem; it only holds for a projective curve X
(see Exercise 16). We give the proof later. We follow the treatment of Tate [58], adopting some technical ideas from Arbarello, De Concini and Kac [3]. 7.4 Linear Algebra in Inﬁnite Dimensional Vector Spaces We will be interested in the inﬁnite dimensional analogues of certain notions for ﬁnite dimensional vector spaces that we now recall. The trace of a square n × n matrix A = (aij ) is the sum of its diagonal elements: Tr A = aii. The trace equals the coefﬁcient in t n−1 of the characteristic polynomial det(A + tE). It follows at once that for a nondegenerate matrix C we have C−1AC Tr = Tr C. (3.90) Thus for a linear map ϕ : L → L of a ﬁnite dimensional vector space L over a ﬁeld k, the traces of the matrix of ϕ with respect to different bases coincide; the common value is called the trace of ϕ and denoted Tr ϕ. The reader interested in a more intrinsic deﬁnition of Tr ϕ may consult Exercises 17 and 18. If we need to emphasise the role of the vector space L we write TrL ϕ in place of Tr ϕ. The function Tr ϕ has the following obvious properties. It is a linear function on the space E(L) of all linear endomorphisms L → L, that is, Tr(αϕ + βψ) = α Tr ϕ + β Tr ψ for α, β ∈ k and ϕ, ψ ∈ E(L). (3.91) In addition, the relation (3.90) can be rewritten as Tr(AC) = Tr(CA) for a nondegenerate matrix C. But both sides of this equality are polynomial functions of the entries of A and C, and since they coincide on the dense subset det C = 0, they coincide everywhere. Thus the trace has the property Tr(ϕψ) = Tr(ψϕ). (3.92) Finally, if M ⊂ L is a vector subspace invariant under ϕ then ϕ deﬁnes a linear endomorphism of M and of the quotient L/M, and L Tr ϕ = Tr
M ϕ + Tr L/M ϕ. (3.93) We now consider the case that L is not necessarily ﬁnite dimensional (and we are speciﬁcally interested in the inﬁnite dimensional case). The trace can then only be deﬁned for special types of linear maps. In the ﬁrst instance, these are the linear maps ϕ whose image subspace ϕ(L) is ﬁnite dimensional. This is an algebraic 220 3 Divisors and Differential Forms analogue of the notion of compact operator (or completely continuous operator) in functional analysis. For such a map, the image ϕ(L) is a ﬁnite dimensional vector subspace of L that is invariant under ϕ. For any ﬁnite dimensional ϕ-invariant subspace V the trace TrV ϕ is well deﬁned, and an obvious veriﬁcation shows that TrV ϕ = TrW ϕ for any two ﬁnite dimensional ϕ-invariant subspaces V and W that both contain ϕ(L). This common value is independent of the choice of V, and we take it as the deﬁnition of Tr ϕ; one checks easily that for this class of maps, the trace Tr ϕ satisﬁes properties (3.91), (3.92) and (3.93) (compare Exercise 19). In what follows, given two vector subspaces A, B ⊂ L, we write A B (and say “A is almost contained in B”) to mean that there exists a ﬁnite dimensional vector subspace W ⊂ L such that A ⊂ B + W ; another way to express the same condition is to say that (A + B)/B is ﬁnite dimensional. In these terms, we can write the above condition on the ﬁnite dimensionality of the image ϕ(L) as ϕ(L) {0}. We now go on to a slightly more involved situation. Suppose given a ﬁxed vector subspace M ⊂ L. Write E(L, M) for the set of linear maps ϕ : L → L such that ϕ(L) M and ϕ(M) {
0}. It is obvious that E(L, M) is a ring. Any map ϕ ∈ E(L, M) satisﬁes ϕ2(L) ϕ(M) {0}, so that ϕ2(L) is ﬁnite dimensional; now ϕ2(L) is obviously invariant under ϕ. To maintain condition (3.93), we have no choice other than to set TrL ϕ = Trϕ2(L) ϕ; to see this, it is enough to apply (3.93) to the pair L, ϕ(L), and then to ϕ(L), ϕ2(L). Conversely, for any ϕ-invariant ﬁnite dimensional vector subspace W containing ϕ2(L), if we set Tr ϕ = TrW ϕ then one checks easily that this number is independent of the choice of W, and a trivial veriﬁcation establishes the relation (3.93). We prove that Tr ϕ also satisﬁes (3.91) and (3.92). It is easy to verify that if ϕ, ψ ∈ E(L, M) then not only are ϕ2(L) and ψ 2(L) ﬁnite dimensional, but so are ϕψ(L) and ψϕ(L). Now set V = ϕ2(L) + ϕψ(L) + ψϕ(L) + ψ 2(L). Then V is obviously invariant under both ϕ and ψ, and both Tr ϕ and Tr ψ can be computed by restricting the linear maps to V, so that checking (3.91) and (3.92) reduces to the ﬁnite dimensional case. Remark 3.4 These arguments have obvious generalisations. For example, rather than just one subspace M, we could consider a nested pair M1 ⊃ M2, and the set E(L, M1, M2) of linear maps ϕ such that ϕ(L) M1, ϕ(M1) M2, ϕ(M2) {0}. Then ϕ3(L) is ﬁnite dimensional and ϕ-invariant and we can deﬁne Tr ϕ
as the trace of ϕ restricted to any ﬁnite dimensional ϕ-invariant subspace V ⊃ ϕ3(L). To prove the analogue of relations (3.91) and (3.92) we should set 8 V = i=1 fi(L) where the fi are the composites of three of ϕ and ψ, that is ϕ3, ϕ2ψ, ϕψϕ, ψϕ2, ϕψ 2, ψϕψ, σ 2ϕ, ψ 3. One obviously has E(L, Mi) ⊂ E(L, M1, M2) for i = 1, 2 and for maps ϕ ∈ E(L, M1) or ϕ ∈ E(L, M2) the two resulting deﬁnitions of trace agree. Compare also Exercise 20. Remark 3.5 Property (3.92) holds under a weaker assumption, namely, assuming only that ϕ(M) M, or the same for ψ. We write E(L, M) for the set of such 7 The Riemann–Roch Theorem on Curves 221 maps. It is obvious that E(L, M) is a ring, and that E(L, M) is contained in it as a two-sided ideal. Suppose now that ϕ ∈ E(L, M) and ψ ∈ E(L, M). A tautological veriﬁcation shows that the spaces N1 = ψϕψ(L) and N2 = (ϕψ)2(L) are ﬁnite dimensional, with ϕ(N1) ⊂ N2, ψ(N2) ⊂ N1, N1 is invariant under ψϕ and N2 is invariant under ϕψ. Then Tr ϕψ = TrN2 ϕψ whereas Tr ψϕ = TrN1 ψϕ, and it follows from the corresponding ﬁnite dimensional result (see Exercise 19) that from which it follows that N1 ψϕ = Tr N2 ϕψ, Tr Tr ψϕ = Tr ϕψ. In what follows we consider maps ϕ, ψ ∈ E
(L, M). We separate the assumptions deﬁning the set E(L, M) into two part, deﬁning E1(L, M) as the set of all maps ϕ such that ϕ(L) M and E2(L, M) as the set of all such that ϕ(M) 0. Then by deﬁnition E1(L, M) ∩ E2(L, M) = E(L, M). A tautological veriﬁcation shows that both the Ei(L, M) for i = 1, 2 are contained in E(L, M) and are double sided ideals in it. Lemma We have the equality E(L, M) = E1(L, M) + E2(L, M). For the proof we need to choose some projector of L onto M, that is, a linear operator π such that π(L) = M and π 2 = π (see Exercise 21). For ϕ ∈ E(L, M), we have a decomposition ϕ = ϕ1 + ϕ2 where ϕ1 = πϕ and ϕ2 = (1 − π)ϕ. An obvious veriﬁcation then shows that ϕ1 ∈ E1(L, M) and ϕ2 ∈ E2(L, M). The decomposition of ϕ ∈ E(L, M) as a sum ϕ = ϕ1 +ϕ2 is of course not unique. ϕ1, ϕ2 are deﬁned only up to replacing (ϕ1, ϕ2) → (ϕ1 + ξ, ϕ2 − ξ ) for some ξ ∈ E(L, M). In what follows we use the notation [ϕ, ψ] for the commutator [ϕ, ψ] = ϕψ − ψϕ of elements ϕ, ψ of any ring. Theorem-Deﬁnition If ϕ, ψ ∈ E(L, M) commute (that is ϕψ = ψϕ) and ϕ1, ψ1 are their components in the decomposition of the above Lemma, then [ϕ1, ψ1] �
� E(L, M) and Tr[ϕ1, ψ1] is an element of k independent of the choice of decomposition of ϕ and ψ. We denote it by ϕ, ψ. It is obvious that [ϕ1, ψ1] ∈ E1(L, M) and that [ϕ1, ψ1] ≡ [ϕ, ψ] ≡ 0 mod E2(L, M), that is, [ϕ1, ψ1] ∈ E2(L, M). Because of this, the effect of replacing ϕ1 → ϕ1 + ξ changes Tr[ϕ1, ψ1] by adding the expression Tr[ξ, ψ1], which is 0 by Remark 3.5 following the proof of property (3.92). 222 3 Divisors and Differential Forms Since Tr[ϕ1, ψ1] is invariant under the change ϕ1 → ϕ1 + ξ (or equally, ψ1 → ψ1 + ξ ), we may replace ψ1 by ψ (or ϕ1 by ϕ) in the preceding deﬁnition, so that, for example ϕ, ψ = Tr[ϕ1, ψ1] = Tr[πϕ, ψ]. (3.94) Our deﬁnition of ϕ, ψ is obviously linear in ϕ and ψ. It has the following remarkable property: ϕ, ψχ = ϕψ, χ + ϕχ, ψ for maps ϕ, ψ, χ ∈ E(L, M). For this it is sufﬁcient that only ϕ and χ commute. (3.95) To prove this, we ﬁnd decompositions of ϕ, ψ and χ as guaranteed by the above Lemma, thus obtaining the corresponding components ϕ1, ψ1 and χ1. To compute ϕ, ψχ, by deﬁnition, we need to ﬁnd the corresponding component (ψχ)1; one checks easily that for this we can take ψ1χ1. In the same way, for the components in the right-hand side of (3.95), we can take
ϕ1ψ1 for (ϕψ)1 and ϕ1χ1 for (ϕχ)1. We now use the identity [α, βγ ] − [αβ, γ ] − [αγ, β] = β, [α, γ ], that holds for any elements of any ring. This identity is equivalent to the well-known Jacobi identify, which means the the map x → [β, x] is a derivation of the Lie algebra with bracket [x, y]. In the proof of the preceding theorem we saw that [ψ1, χ1] ∈ E(L, M). This implies that Tr[ψ1, [ϕ1, χ1]] = 0 (see Remark 3.5 following the proof of property (3.92)). This proves the relation (3.95). The relation (3.95) just established already hints at connections between the construction under consideration and the notion of differentials. Namely, in Section 5.2 we deﬁned the module of differentials ΩA for any commutative ring A (as an algebra over a subring A0, that we take to be the ﬁeld k in what follows). We saw that ΩA is generated as an A-module by dt for t ∈ A. Thus it is generated as an Abelian group by udt for u, t ∈ A. The relations between these generators are all obtained from the relations (3.57) by multiplying by u ∈ A. Thus they are of the form ud(f + g) = udf + udg and udfg = uf dg + ugdf for u, f, g ∈ A. (3.96) We must add to these the relations expressing that ΩA is an A-module, namely (u + v)df = udf + vdf. (3.97) It follows that any function F (u, f ) ∈ k of a u, f ∈ A vanishing on the subgroup generated by the relations (3.96)–(3.97) deﬁnes a linear function on ΩA. Suppose in particular that the ring A acts on a vector space L in such a way that the product of elements of A acts as the composition of linear maps, and such that each
f ∈ A deﬁnes a linear map f ∈ E(L, M) (for a subspace M ⊂ L that is independent of f ∈ A. Then the function f, g deﬁnes a linear function r : ΩA → k such that f, g = r(f dg). 7 The Riemann–Roch Theorem on Curves 223 Example 3.17 Set L = k((t)) and let A act by multiplication on itself; we take M = k[[t]]. It is obvious that f ∈ E(L, M) for any f ∈ A. By what we said above, there exists a linear function r : ΩA → k such that r(f dg) = f, g. We determine this function r. Since every differential f dg can be written in the form udt, it is enough to determine the function u, t. For this, we deﬁne the projector π : L → M by the condition π(t k) = 0 for k < 0 and π(x) = x for x ∈ M = k[[t]]. We verify directly that if we set ϕ1 = πϕ for any ϕ = u ∈ M = k[[t]], and ψ1 = πψ with ψ = t, then the map a = [ϕ1, ψ1] satisﬁes aL ⊂ M and aM = 0, so that Tr a = 0, that is, u, t = 0. Thus it only remains to determine t −n, t for n > 0. The corresponding map a = [ϕ1, ψ1] is calculated by a direct substitution. We obtain Thus obviously t k a = * 0 1 * Tr a = if k = n − 1; if k = n − 1. 0 if n = 1; 1 if n = 1. This shows that u, t = a−1 is the coefﬁcient of t −1 in the Laurent series expansion of u. Thus Resx(f dg) = f, g. This is the intrinsic deﬁnition of the residue at a point. Example 3.18 We now set L = A = k(X), let A act on itself by multiplication, and set M = Ox for some
point x ∈ X. We are in a situation close to that of Example 3.17. Set L = k((t)) and M = k[[t]]; then for f, g ∈ k(X) we can compute the expression f, g assuming either that f, g ∈ k(X), or f, g ∈ k((t)). Let us check that we get one and the same result. We are talking about comparing the expressions for ϕ, ψ for different spaces L and M. So we take account of them in the notation, writing ϕ, ψL M for given L and M. We have the following inclusions of subspaces: L ⊃ M L ⊃ M The result we need is a consequence of the following fact. Lemma Assume the above inclusions, and let ϕ, ψ : L → L be linear maps that take the subspace L to itself. Then ϕ, ψL M = ϕ, ψL M. The proof follows as a tautology from the fact that we can choose the projectors π : L → M and π : L → M in a compatible way, that is, so that π \|L = π. For the possibility of making this choice, see Exercise 21. 224 3 Divisors and Differential Forms 7.5 The Residue Theorem We go back to the deﬁnition of trace and to the expression ϕ, ψ introduced in Section 7.4, to study their dependence on the choice of the subspace M. To underline this dependence, we write ϕ, ψM for ϕ, ψ ∈ E(L, M). Suppose now that we are given two subspaces M, N ⊂ L, and that ϕ and ψ are contained both in E(L, M) and in E(L, N ). It is obvious that they are then also contained in E(L, M + N ) and in E(L, M ∩ N ). Theorem 3.26 We have the relation ϕ, ψM + ϕ, ψN = ϕ, ψM+N + ϕ, ψM∩N. (3.98) One proves easily that we can choose the four projectors πM, πN, πM+N and πM�
�N to satisfy πM + πN = πM+N + πM∩N (compare Exercise 21). It might seem that the relation (3.98) follows from formulas (3.94) and (3.91). But actually, we are dealing with maps that belong to E(L, M) for different M, and the relation (3.85) is not applicable. However, we can use Remark 3.4. If is clear that [πMϕ, ψ] and [πM+N ϕ, ψ] are contained in the set E(L, M + N, M), and hence by the choice of πM, πN, πM+N and πM∩N, we have ϕ, ψM − ϕ, ψM+N = Tr (πM − πM+N )ϕ, ψ (πM∩N − πN )ϕ, ψ = Tr. Exactly the same argument together with (3.95) shows that the last expression equals ϕ, ψM∩N − ϕ, ψN. Theorem 3.27 (The Residue Theorem) For a nonsingular projective curve X and a rational 1-form ω ∈ Ω 1(X) the residue Resx ω can only be nonzero at a ﬁnite set of points x ∈ X, and we have the equality x∈X Resx ω = 0. We write ω in the form f dg for rational functions f, g ∈ k(X). Write L = R for the space of distributions on X (see Section 7.2), and consider the action of k(X) on L by multiplication. Set M = R(0) (see Section 7.2) and N = k(X). It follows from the deﬁnition of distribution that f ∈ E(R, R(0)) for any f ∈ k(X), and a fortiori that f ∈ E(R, k(X)). We apply Theorem 3.26 to any two functions f, g ∈ k(X). We get that f, gR(0) + f, gk(X) = f, gR(0)+k(X) + f,
gR(0)∩k(X). (3.99) By Theorem 3.24 the space R(0) + k(X) has ﬁnite codimension in R, and the intersection R(0) ∩ k(X) = k, and is thus ﬁnite dimensional. But if either M is ﬁnite dimensional (that is, M 0), or M has ﬁnite codimension in L (that is, L M) then f, gM = 0 for any f, g. In the ﬁrst case πMf and πMg are linear 7 The Riemann–Roch Theorem on Curves 225 transformations of a ﬁnite dimensional subspace of M, on which we can calculate the trace; the assertion then follows from (3.91). In the second case (1 − π)L is ﬁnite dimensional, which reduces us to the ﬁrst case. Finally, if M is invariant under ϕ and ψ then we have πk(X)f = f, [πk(X)f, πk(X)g] = [f, g] = 0 and ϕ, ψM = 0 holds trivially. Thus (3.99) shows that and it only remains to convince ourselves that f, gR(0) = 0, f, gR(0) = x∈X Resx(f dg). Write S for the ﬁnite set of points that are poles of either f or g, and P for the set of maps taking each s ∈ S to a function fs ∈ k(X). Let N be the space R(0) for the curve X \ S. Then R(0) = N + Q, where Q consists of the distributions that takes each s ∈ S into a function fs ∈ Os (that is, fs is regular at s). Theorem 3.26 show that f, gR(0) = f, gN + f, gQ. However by construction, N is invariant under multiplication by f and g, and hence Os, where f, gN = 0. It remains to compute f, gQ. We can write Q as each summand Os corresponds to the point s ∈ S. Applying Theorem 3.
26 again (applied to an arbitrary number of components), we get that s∈S ) f, gQ = f, gOs. Here Os is the set of all distributions for which rs is any function in Os, and rt = 0 for t = s. It remains only to apply the Lemma at the end of Section 7.4, and = Ress(f dg). This completes the proof using Example 3.18, we get that f, gOs of Theorem 3.27. 7.6 The Duality Theorem We saw in Section 7.3 that the Riemann–Roch Theorem would follow from Theorem 3.25 once we prove that the pairing between L(K − D) and R/(R(D) + K) deﬁned by (3.92) is nondegenerate. There we also proved that this pairing is only well deﬁned on the basis of the Residue Theorem that we have just proved (Theorem 3.27). Now that we know it is well deﬁned, we can proceed to the proof of nondegeneracy. 226 3 Divisors and Differential Forms Theorem 3.28 The pairing (r, ω) between the two spaces Ω 1(D) = L(K − D) and Λ(D) = (R/R(D) + k(X)) deﬁned by (r, ω) = x Res(rx, ω) is nondegenerate. The proof of the theorem breaks up into two parts: (a) nondegeneracy in the second argument ω; and (b) nondegeneracy in the ﬁrst argument x. (a) Nondegeneracy in the First Argument. The statement for ω ∈ Ω 1(D) is that if (r, ω) = 0 for every r ∈ R/(R/R(D) + k(X)) then ω = 0. Since we already know that (R(D) + k(X), ω) = 0, our assertion is that for ω ∈ Ω 1(X), the equality (r, x) = 0 for every r ∈ R implies that ω = 0. Suppose that ω = 0 and let x be a point appearing in div ω with coef�
�cient n. Then we can write ω in the form f dt, where t is some local parameter at the point x, and vx(f ) = n. Consider the distribution r given by rx = t −n−1 and ry = 0 for y = x. Then obviously (r, ω) = Resx rxω = 0. (b) Nondegeneracy in the Second Argument. This can also be restated as saying that taking a form ω ∈ Ω 1(D) to the linear function r → (r, ω) on R deﬁnes a surjective linear map of Ω(D) onto the whole space Λ(D)∗. This is a more delicate fact. For the proof, consider the set Λ of all linear functions on R/k(X) that vanish on some space R(D). Since multiplication by a rational function f ∈ k(X) is deﬁned on R and on k(X), it follows that R/k(X) is a module over k(X). We prove that it is one dimensional. Indeed, any rational differential 1-form ω ∈ Ω 1(X) with ω = 0 deﬁnes, as we have seen, a nonzero function ϕ ∈ Λ. Let us prove that any two functions ϕ, ψ ∈ Λ are linearly dependent over k(X). We can assume that both ϕ, ψ ∈ R(D) for some divisor D. We consider some sufﬁciently large natural number n # 0 and functions f, g ∈ L(nP ). Under the assumption that ϕ and ψ are linearly independent, the combinations f ϕ + gψ are all different and give a space of dimension 2(nP ), which is obviously contained in Λ(D − nP ). Therefore 2(nP ) ≤ λ(D − nP ). This inequality gives a contradiction for large n. We use Theorem 3.25, which says that (D) − λ(D) = deg D + c, where c is some constant, not depending on D. If we choose n > deg D, so that (D − nP ) = 0. Then λ(D − nP ) = n + c, where
c is constant (for a ﬁxed D). But (nP ) ≥ n + c. Taking ϕ to be the linear form deﬁned by a differential form ω, we deduce that any linear form ϕ ∈ Λ is deﬁned by a differential form f ω. It remains to prove that if the linear form (r, ω) vanishes on R(D) then ω ∈ Ω 1(D). The proof at this point is the same as that of Assertion (a). Suppose that ω /∈ Ω 1(D). Then there exists a point x appearing in the divisor D with multiplicity n, and a local parameter t at x such that ω = f dt and vx(f ) < n. Consider the distribution r for which rx = t −vx (f )−1 and ry = 0 for y = x. Then obviously r ∈ R(D) but (r, ω) = 0. 7 The Riemann–Roch Theorem on Curves 227 7.7 Exercises to Sections 6–7 1 Prove that an element f ∈ k(X) satisﬁes df = 0 if and only if f ∈ k (in the case of a ﬁeld of characteristic 0), or f = gp (in the case char k = p > 0). [Hint: Use Theorem 3.21 and the following lemma: if L ⊂ K is a ﬁnite separable ﬁeld extension in characteristic p > 0, and x ∈ K has the property that its minimal polynomial is of the form ap i xi with ai ∈ L then x = yp with y ∈ K.] 2 Let X and Y be nonsingular projective curves and ϕ : X → Y a regular map such that ϕ(X) = Y and x ∈ X, y = f (x) ∈ Y, and let t be a local parameter on Y at y. Prove that the number ex = vx(f ∗(dt)) does not depend on the choice of the local parameter t and that ex > 0 if and only if x is a branch point of ϕ. The number ex is the ramiﬁcation multiplicity of ϕ at x
. (Compare Section 3.1, Chapter 7.) 3 In the notation of Exercise 2, suppose that ϕ∗(y) = lixi where y is a divisor consisting of the single point y. Suppose that the characteristic of k is equal either to 0, or to a prime p > li. Prove that exi = li − 1. 4 In the notation of Exercises 2–3, suppose that Y = P1. Prove that g(X) is given by 2g(X) − 2 = −2 deg ϕ + x∈X ex (the Hurwitz ramiﬁcation formula). Generalise this relation to the case of Y an arbitrary curve. 5 Suppose that ϕ : X → Y satisﬁes the conditions of Exercise 2. Prove that a rational differential ω ∈ Ω 1(Y ) is regular if and only if ϕ∗(ω) ∈ Ω 1[X]. 6 Let L be an n-dimensional vector space. Write Ψm for the set of all functions ψ of mn vectors xij ∈ L for i = 1,..., m and j = 1,..., n that satisfy the conditions: (a) ψ is linear in each argument; (b) ψ is skewsymmetric as a function of xi0j, for any ﬁxed i0 and j = 1,..., n; (c) ψ is symmetric as a function of xij0, for any ﬁxed j0 and i = 1,..., m. Suppose that char k > m. Prove that every function ψ ∈ Ψ is determined by its values ψy1...yn at vectors xij = yj, and that ψy1...yn = d mψe1...en, where d is the determinant of the coordinates of the vectors y1,..., yn in the basis e1,..., en. Suppose that ξ1,..., ξn ∈ L∗ = (det \|ξi(yj )\|)m is written is the dual basis. The function ψ for which ψy1...yn (ξ1 ∧ · · · ∧ ξn)m. Prove that Ψ
m is one dimensional and is based by (ξ1 ∧ · · · ∧ ξn)m. r Θ ∗ 7 Generalise the construction of regular and rational differential n-forms, replacing x by Ψm throughout. The resulting object is called a differential form the space of weight m. Prove that in the analogue of (3.63) we should replace J by J m. Prove that a differential form of weight m has a divisor, that all these divisors belong to one divisor class, and that this class is mK. Generalise Theorem 3.22. 228 3 Divisors and Differential Forms 8 Compute the space of regular differential forms of weight 2 on a hyperelliptic curve. [Hint: Write them in the form f (dx)2/y2.] 9 Verify formula (3.89) by direct calculations in the case X = P1. [Hint: Set ω = f dx and write the rational function f (x) ∈ k(P1) as a fraction in simplest form.] Convince yourself that the formula is false for X = A1. 10 Let L be a ﬁnite dimensional vector space. A linear transformation ϕ of L has rank 1 if its image is 1-dimensional. If a = 0 is in the image of ϕ then ϕ is of the form ϕ(x) = χ(x) · a for x ∈ L, where χ ∈ L∗ is some linear form on L. We write ϕ = P χ a for such a map. Prove that linear transformations of the form P χ a generate the whole space of linear transformations of L. Verify that P χ1+χ2 a − P χ1 a − P χ2 a = 0 and P χ a1+a2 − P χ a1 − P χ a2 = 0. Prove that any linear relation between the transformations P χ a follow from these. (In other words, this says that the vector space of linear transformations of L is isomorphic to the tensor product L ⊗ L∗.) 11 Prove that the linear function λ(P χ a ) = χ(a) vanishes on the relations of Exercise 10. Deduce from this that λ has a unique extension to the vector space of linear transformation of
L, and that its value on any linear transformation ϕ equals Tr ϕ. 12 Let L1 and L2 be ﬁnite dimensional vector spaces and ϕ : L1 → L2 and ψ : L2 → L1 two linear transformations. Prove that Tr(ϕψ) = Tr(ψϕ). 13 Suppose given a vector space L and k nested subspaces L ⊂ M1 ⊂ · · · ⊂ Mk. Write E(L, M1,..., Mk) for the set of all linear transformations ϕ of L such that ϕ(L) M1, ϕ(M1) M2,..., ϕ(M1) (0). Prove that for any such ϕ, the space ϕk(L) is ﬁnite dimensional and ϕ-invariant. If V ⊃ ϕk(L) is any ﬁnite dimensional ϕ-invariant vector subspace then TrV ϕ is independent of the choice of V. Prove that under these assumptions, the relations (3.91) and (3.92) hold. 14 Prove that constructing a projector π : L → M is equivalent to ﬁnding a complement to M in L, that is, a vector subspace M such that L = M ⊕ M. Under this, π(M) = 0. Prove that such a complement M always exists, and deduce the existence of a projector π. [Hint: The construction of M requires an appeal to Zorn’s Lemma.] 15 Suppose that L ⊃ M, L ⊃ L, M ⊃ M and L ⊃ M. Prove that there exist projectors π : L → M and π : L → M that are compatible, in the sense that π = π on L. 8 Higher Dimensional Generalisations 229 16 Verify the relation deg K = 2g − 2 for hyperelliptic curves and nonsingular curves in the plane. 17 Prove that for a hyperelliptic curve, the ratio between regular differential forms generate a subﬁeld of k(X) isomorphic to the ﬁeld of rational functions. From this deduce that a nonsingular plane curve Xm ⊂ P2
of degree m > 3 is not hyperelliptic. 18 Prove that for a hyperelliptic curve, the rational map corresponding to the canonical class is not an embedding. 19 Prove that if the map corresponding to the canonical class of a curve X is not an embedding then X is rational or hyperelliptic. [Hint: If one or other of the conditions (3.81) fails then the Riemann–Roch theorem gives (x) ≥ 2 or (x + y) ≥ 2.] 20 Prove that a nonsingular cubic 3-fold X3 ⊂ P4 is unirational. [Hint: Use Theorem 1.28 to show that X contains a line l. Using Exercise 8 of Section 5.5, prove that there exists an open set U ⊂ X with U ∩ l = ∅ such that the tangent ﬁbre space to U is isomorphic to U × A3. Write P2 for the projective plane consisting of lines through the origin of A3. For a point ξ = (u, α) with u ∈ l ∩ U and α ∈ P2, denote by ϕ(ξ ) the point of intersection of the line α lying in ΘX,u with X. Prove that ϕ deﬁnes a rational map P1 × P2 → X.] 21 Let o be a point of an algebraic curve X of genus g. Using the Riemann–Roch theorem, prove that any divisor D with deg D = 0 is equivalent to a divisor of the form D0 − go, where D0 > 0, deg D0 = g. This is a generalisation of Theorem 3.10. 22 Let X ⊂ P2 be an irreducible nonsingular plane curve with equation F = 0, and suppose that α = (α0 : α1 : α3) /∈ X and x ∈ X. The multiplicity cx of x in the 2 i=0 ∂F /∂xi is called the multiplicity of tangency at x. Prove divisor of the form that cx = ex is the ramiﬁcation multiplicity of x with respect to the map ϕ : X → P1 given by projecting from α. Deduce that c = x∈X cx is the number of tangent lines to
X through α, counted with multiplicities. It does not depend on α. It is called the class of X. Prove that c = n(n − 1) where n = deg X. 23 Prove that if X is a nonsingular afﬁne hypersurface then KX = 0. 8 Higher Dimensional Generalisations We discuss here informally how the facts proved in the preceding section for algebraic curves generalise to irreducible projective algebraic surfaces. We do not provide any proofs. The reader can ﬁnd them in Shafarevich [69], Bombieri and Husemoller [14] or Barth, Peters and Van de Ven [9]. In addition, we restrict ourself to the case of a ﬁeld of characteristic zero. 230 3 Divisors and Differential Forms The analogue of curves of genus >1 are surfaces for which some multiple of the canonical class deﬁnes a birational embedding. These are called surfaces of general type, and for them, the birational classiﬁcation reduces in a certain sense to the projective classiﬁcation. The fundamental result for surfaces of general type is that already 5K deﬁnes a regular map that is a birational embedding. It remains to enumerate the surface not of general type. These play the role of curves of genus 0 and 1, and are given by analogous constructions. The analogue of rational curves are, in the ﬁrst place, the rational surfaces, that is, surfaces birationally equivalent to P2, and then the ruled surfaces; these are the surfaces that can be mapped to a curve C such that all the nonsingular ﬁbres are isomorphic to the projective line P1. That is, they are algebraic families of lines. There are three types of surfaces that play the role of curves of genus 1. The ﬁrst of these are the Abelian surfaces, that is, 2-dimensional Abelian varieties. The second type of surfaces, called K3 surfaces, share the property of Abelian surfaces that their canonical class is zero. However, in distinction to Abelian surfaces, they have no regular differential 1-forms, whereas according to Proposition of Section 6.2, Abelian varieties have invariant 1-forms, that are therefore regular. The third type are the elliptic surfaces, that is, families of ellipt
ic curves. These surfaces have a map f : X → C to a curve C such that over every y ∈ C for which the ﬁbre f −1(y) is a nonsingular curve (that is, for all but ﬁnitely many y), this curve has genus 1. The main theorem asserts that all the surfaces that are not of general type are exhausted by the ﬁve types just listed: rational, ruled, Abelian, K3 and elliptic surfaces. To get a better idea of these classes of surfaces, it is convenient to classify them by the invariant κ, the maximal dimension of the image of X under the rational map given by a divisor class nK for n = 1, 2,.... If (nK) = 0 for all n then there are no such maps, and we12 set κ = −∞. Here is the result of the classiﬁcation. The surfaces of general type are those with κ = 2. Surfaces with κ = 1 are all elliptic surfaces; to be more precise, these are the elliptic surfaces for which nK = 0 for n = 0. For an elliptic surface X, the order of the canonical class in Cl X is either inﬁnite or a divisor of 12. The surfaces with κ = 0 are characterised by the condition 12K = 0. These are thus the elliptic surfaces with 12K = 0, together with K3 surfaces and 2-dimensional Abelian varieties. Surfaces with κ = −∞ are the rational or ruled surfaces. For each of these ﬁve types of surfaces there is a characterisation in terms of invariants, in the same way that the equality g = 0 characterises rational curves. We only give such a characterisation for the two ﬁrst types. For this we use the result of Exercise 7, according to which the numbers (mK) for m ≥ 0 are birational invariants of a nonsingular projective variety. These are called the plurigenera,, and denoted by Pm = (mK). In particular P1 = hn = dim Ω n[X], where n = dim X. 12κ = −1 also occurs in the literature. The invariant κ is usually called the Kodaira dimension, although it was introduced in different contexts by the Shafare
vich seminar [69] and by Iitaka. 8 Higher Dimensional Generalisations 231 Rationality Criterion A surface X is rational if and only if Ω 1[X] = 0 and P1 = P2 = 0. The positive solution of the Lüroth problem (discussed in Section 6.4) for surfaces over an algebraically closed ﬁeld of characteristic 0 follows at once from this criterion. Ruledness Criterion A surface X is ruled if and only if P3 = P4 = 0. Generalisations of the results of this section to varieties of dimension ≥3 are not known, although there has been a lot of progress on this question in recent years. For this see for example the surveys Esnault [26], Kawamata [45] and Wilson [80], and for the relation with minimal models, Kawamata, Matsuda and Matsuki [46]. We discuss brieﬂy the analogous questions for algebraic varieties of arbitrary dimension n. The basic invariant of a variety X is its so-called canonical dimension or Kodaira dimension κ(X). By analogy with the case of curves and surfaces, this is deﬁned as the upper bound of the dimensions of the images of X under the pluricanonical maps deﬁned by the classes mKX for natural number m. If (mKX) = 0 for all m > 0 we set κ(X) = −∞. Thus κ(X) may take the value −∞, 0, 1,..., n. If κ(X) = n we say that X is a variety of general type. In the case of curves, those of general type are those of genus g ≥ 2. For surfaces, those not of general type are in some sense exceptional and can be described as just discussed. This justiﬁes the term “of general type”. ) For an arbitrary variety X, all the spaces L(mKX) for m ≥ 0 ﬁt together into a L(mKX). In R, the multiplication of elements of the spaces single ring R = L(mKX) is deﬁned starting from the condition: if f ∈ L(pKX) and g ∈ L(qKX) then fg ∈ L((p + q)KX). A ring with this property
is called a graded ring. The ring R deﬁned above is called the canonical ring of X. m≥0 We make the following assumptions on the ring R: (a) R is generated as an algebra over k by a ﬁnite number of generators. (b) R is generated by its elements of degree 1, that is, L(KX). Assuming (a), we can arrange for an analogue of condition (b) to hold by a simple modiﬁcation of the statement. Namely, for any natural number r we set R(r) = ) L(rmKX). A simple algebraic argument then shows that R(r) is generated by its elements of degree 1, that is by L(rKX), provided that r is sufﬁciently large and divisible. m≥0 In contrast, the question of whether (a) holds is very difﬁcult, and is at present unsolved in complete generality. If R satisﬁes (a) and (b) and N + 1 = dim L(rKX) then R is the surjective image of a homomorphism ϕ : k[x0,..., xN ] → R. One sees easily that its kernel is a homogeneous ideal, and hence R deﬁnes a cone over a projective variety Y, and the homomorphism ϕ deﬁnes a map X → Y. To be able to guarantee condition (b), we apply this construction to the ring R(r) for sufﬁciently large and divisible r. One 232 3 Divisors and Differential Forms can show that the result does not depend on the choice of r. Thus (assuming that condition (a) holds), we obtain a certain standard canonical model of a variety X. As the current edition of this book was under preparation, two independent proof of the following fundamental result were announced, based on different ideas (see Birkar, Cascini, Hacon and McKernan [11] and Siu [73]): For a variety of general type, the canonical ring R is ﬁnitely generated. Chapter 4 Intersection Numbers 1 Deﬁnition and Basic Properties 1.1 Deﬁnition of Intersection Number The theorems proved in Section 6.2, Chapter 1 on the dimension
of intersection of varieties often allow us to assert that some system of equations has solutions. However, they do not say anything about the number of solutions if this number is ﬁnite. The distinction is the same as that between the theorem that roots of a polynomial exist, and the theorem that the number of roots of a polynomial equals its degree. The latter result is only true if we count each root with its multiplicity. In the same way, to state general theorems on the number of points of intersection of varieties, we must assign certain intersection multiplicities to these points. This will be done in the present section. We will consider intersection of codimension 1 subvarieties on a nonsingular variety X. We are interested in the case that the number of points of intersection is ﬁnite. If dim X = n and C1,..., Ck are codimension 1 subvarieties with nonempty intersection, then by Theorem 1.22 and Corollary 1.7, we have dim(C1 ∩ · · · ∩ Ck) > 0 if k < n. Hence it is natural to consider the case k = n. The theory that we apply in the following is simpler if we consider arbitrary divisors in place of codimension 1 subvarieties. Thus we consider n divisors D1,..., Dn on an n-dimensional variety X. If x ∈ X with x ∈ Supp Di = 0 then we say that Supp Di and dimx D1,..., Dn are in general position at x. The condition means that in some neigh- bourhood of x, the intersection Supp Di consists of x only. If D1,..., Dn are in Supp Di then this subvariety either general position at all points of the subvariety consists of a ﬁnite number of points, or is empty. We then say that D1,..., Dn are in general position. We deﬁne intersection numbers ﬁrst of all for effective divisors in general position. Suppose that D1,..., Dn are effective and in general position at x, and have local equations f1,..., fn in some neighbourhood of x. Then there exists a neighbourhood U of x in which f1,..., fn are regular and have
no common zeros on U other than x. It follows from the Nullstellensatz that the ideal generated by f1,..., fn in I.R. Shafarevich, Basic Algebraic Geometry 1, DOI 10.1007/978-3-642-37956-7_4, © Springer-Verlag Berlin Heidelberg 2013 233 234 4 Intersection Numbers the local ring Ox of x contains some power of the maximal ideal mx. Suppose that (f1,..., fn) ⊃ mk x. (4.1) We consider the quotient Ox/(f1,..., fn) as a vector space over k; it is ﬁnite dimensional. Indeed, in view of (4.1), for this it is enough to prove that x < ∞. This last condition follows at once from the theorem on power dimk Ox/mk series expansion (Section 2.2, Chapter 2): dimk Ox/mk x equals the dimension of the space of polynomials of degree < k in n variables. From now on we write (E) for the dimension of a k-vector space E. Deﬁnition 4.1 If D1,..., Dn are effective divisors on an n-dimensional nonsingular variety X, in general position at a point x ∈ X, and having local equations f1,..., fn in some neighbourhood of x, then the number Ox/(f1,..., fn) (4.2) is the intersection multiplicity or local intersection number of D1,..., Dn at x. We denote it by (D1 · · · Dn)x. 1,..., f the choice of local equations f1,..., fn: if f f i The number (4.2) actually only depends on the divisors D1,..., Dn and not on n are other local equations then 1,..., f n). = figi with gi a unit of Ox, and hence (f1,..., fn) = (f Now suppose that D1,..., Dn are not necessarily effective divisors. Write in the form Di = D i, with D ≥ 0 having no common compoDi i nents; this expression is unique
. Suppose that D1,..., Dn are in general position at x. Then, since Supp Di = Supp D i, it follows that the divii sors D, D are in general position at x for any permutation ik+1 i1 i1,..., in and any k.,..., D in ∪ Supp D,..., D ik i, D − D i We now deﬁne the intersection number of D1,..., Dn at x by multilinearity, that is, we set (D1 · · · Dn)x = n D i − D i i=1 = n (−1)n−k i1...in k=0 x D i1 · · · D ik D ik+1 · · · D in x. (4.3) Deﬁnition 4.2 If divisors D1,..., Dn on an n-dimensional variety X are in general position, then the number D1 · · · Dn = x∈ Supp Di (D1 · · · Dn)x is called their intersection number. We can formally extend the sum over all points x ∈ X, although of course only the terms with x ∈ Supp Di are nonzero. 1 Deﬁnition and Basic Properties 235 Remark We can also deﬁne intersection numbers without requiring X to be a nonsingular variety; however, we then have to restrict attention to locally principal divisors (Section 1.2, Chapter 3). All the deﬁnitions given above preserve their meaning.13 We now give some examples, with the aim of showing that the deﬁnition of intersection multiplicity just introduced agrees with geometric intuition. Example 4.1 Suppose that dim X = 1, and that t is a local parameter at a point x. Let f be a local equation of a divisor D with vx(f ) = vx(D) = k. Then (D)x = (Ox/(f )) = (Ox/(t k)) = k. Thus in this case the local multiplicity (D)x is just the multiplicity of x in the divisor D. In the following examples we will assume that Di are prime divisors, that is, irreducible codimension 1 sub
varieties of X. Example 4.2 If x ∈ D1 ∩ · · · ∩ Dn then (D1 · · · Dn)x ≥ 1 by deﬁnition. Let us determine when (D1 · · · Dn)x = 1. Now fi ∈ mx, so that (f1,..., fn) ⊂ mx, and since (Ox/mx) = 1, the condition (D1 · · · Dm)x = 1 is equivalent to (f1,..., fn) = mx. In other words, f1,..., fn should form a local system of parameters at x. We saw in Section 2.1, Chapter 2 that this holds if and only if the subvarieties D1,..., Dn intersect transversally at x, that is, x is a nonsingular point on each Di, and ΘDi,x = 0. Example 4.3 Suppose that dim X = 2, and that the point x is nonsingular on both curves D1 and D2. By Example 4.2, (D1D2)x > 1 if and only if the two tangent lines ΘD1,x and ΘD2,x coincide. Let u, v be local parameters at x and f1, f2 local equations of D1, D2, and write fi ≡ αiu+βiv mod m2 x. Then for i = 1, 2, the tangent line ΘDi,x is given by the equation αiξ + βiη = 0, where ξ = dxu and η = dxv are coordinates in ΘX,x. Hence ΘD1,x = ΘD2,x if and only if α2u + β2v = γ (α1u + β1v) for some nonzero γ ∈ k, or in other words, f2 ≡ γf1 mod m2 x. It is thus natural to deﬁne the order of tangency of D1 and D2 at x to be the number k such that there exists an invertible element g ∈ Ox such that f2 ≡ gf1 mod mk+1, and no such g exists for greater values of the exponent k + 1. We now show that the intersection multipl
icity is one plus than the order of tangency of the curves D1 and D2 at x, that is, (D1D2)x = k + 1. x For this note that, because x is a nonsingular point of D1, we can assume that f1 is one element of a system of local parameters at x. On the other hand, g−1f2 is a local equation of D2. Hence we can assume that u, v are local parameters, the local equation of D1 is u, that of D2 is f, and f ≡ u mod mk+1. Then f ≡ u + x 13Although the prime divisors Γi that are components of a Cartier divisor D = ai Γi are not necessarily Cartier, it is still true that any locally principal divisor D can be written D = D − D with D and D effective; this follows in a neighbourhood of any point x ∈ X simply because the rational function ﬁeld k(X) is the ﬁeld of fractions of Ox. 236 4 Intersection Numbers ϕ(u, v) mod mk+2 by u, since otherwise D1 and D2 would have order of tangency > k. Hence, where ϕ is a form of degree k + 1. Moreover, ϕ is not divisible x ϕ(0, v) = cvk+1, with c = 0. (4.4) By deﬁnition of intersection multiplicity, Ox/(u, f ) (D1D2)x =, = Ox/(u) (u, f )/(u). Now obviously, Ox/(u) = O is the local ring of the point x on D1, and the quotient map Ox → O is restriction of functions from X to D1. Moreover, (u, f )/(u) = (f ), where f is the image of f in O. Since, as an element of O, we have f ∈ (mx)k+1 and f ≡ ϕ mod(mx)k+2, and by (4.4) ϕ /∈ (mx)k+2, therefore vx(f ) = k + 1 and (O/(f )) = k + 1. Thus (D1D2)x = k + 1. Example 4.4 Suppose again that dim X =
2, and that the point x is singular on D. This means that f ∈ m2 x, where f is the local equation of D. Hence it is natural to deﬁne the multiplicity of the singularity x ∈ D to be the greatest k such that f ∈ mk x. We prove that for any curve D on X such that D and D are in general position at x, DD ≥ k, x (4.5) and that there exist curves for which (DD)x = k. Let f be a local equation of D. Write O for Ox/mk x and f ∈ O for the image of f. Since f ∈ mk x, we have (DD)x = (Ox/(f, f )) ≥ (O/(f )). By the theorem on power series expansion (Section 2.2, Chapter 2), O is isomorphic to k[u, v]/(u, v)k. Therefore it is isomorphic as a vector space to the space of k+1 polynomials of degree <k in u, v, and has dimension If f ∈ ml then elements of the ideal (f ) correspond to polynomials of x the form f g where g runs through all polynomials of degree ≤k − l. Hence. Since f ∈ m, we have l ≥ 1, and hence ((f )) ≤ 1 + · · · + (k − l) = (O/(f )) = (O) − ((f )) ≥ k. \ ml+1 x k−l+1 2 Now we prove that equality in (4.5) can be achieved. Suppose that f ≡ ϕ(u, v) mod mk+1, where ϕ is a form of degree k. Consider a linear form in u, v not dividing ϕ. At the cost of a linear transformation of u and v we can assume that this is u, with ϕ(0, v) = 0. Take D to be the curve with local equation u. Then (DD)x = (Ox/(u, f )), and, as we have seen in the treatment of Example 4.3, this number equals k. x 1.2 Additivity Theorem 4.1 If D1,..., Dn−1, D at x then D1 · · · Dn−1 + D n D n = x n and D1,..
., Dn−1, D n are in general position D1 · · · Dn−1D n x + D1 · · · Dn−1D n x. (4.6) 1 Deﬁnition and Basic Properties 237 Proof First of all, it is obviously enough to prove Theorem 1 for effective divisors D1,..., Dn−1, D n, D Let f1,..., fn−1, f n. We denote the ring Ox/(f1,..., fn−1) by O, and the images in O of f n. From now on we assume that these divisors are effective. n be local equations of the divisors D1,..., Dn−1, D n, f n, n, f n by D f, g. Then D1 · · · Dn−1D n D1 · · · Dn−1 O/(f ) =, + D D n n and x x D1 · · · Dn−1D n O/(fg) =, x O/(g) =, Since the sequence 0 → (g)/(fg) → O/(fg) → O/(g) → 0 is exact, it follows that O/(fg) = O/(g) + (g)/(fg). (4.7) If g is a non-zerodivisor of O then multiplication by g deﬁnes isomorphisms O ∼= (g) and (f ) ∼= (fg), hence an isomorphism O/(f ) ∼= (g)/(fg), and therefore (g)/(fg) O/(f ). = (4.8) Thus (4.6) follows from (4.7) and (4.8), provided that we can prove that g is a nonzerodivisor of O. A sequence f1,..., fn of n elements of the local ring Ox of a nonsingular point of an n-dimensional variety is called a regular sequence if each fi is a nonzerodivisor of Ox/(f1,..., fi−1) for i = 1,..., n. The arguments just given show that Theorem 4.1 follows from the next asser- tion. Lemma 4.1 If the divisors D1,
..., Dn are in general position at a nonsingular point x, then their local equations f1,..., fn form a regular sequence. In turn, the proof of Lemma 4.1 requires the following simple auxiliary result, which is a general property of local rings proved in Proposition A.13. Lemma 4.2 The property that a sequence of elements is a regular sequence is preserved under permuting the elements of the sequence. Proof of Lemma 4.1 The proof is by induction on the dimension n of X. From the assumptions of the lemma and the theorem on the dimension of intersection (Section 6.2, Chapter 1) it follows that dimx(Supp(D1) ∩ · · · ∩ Supp(Dn−1)) = 1. Hence we can ﬁnd a function u such that u(x) = 0, x is a nonsingular point of V (u) and the n divisors D1,..., Dn−1, div u are in general position at x. For this, we need only take u to be the equation of a hyperplane through x not containing 238 4 Intersection Numbers ΘX,x or any component of the curve Supp(D1) ∩ · · · ∩ Supp(Dn−1). Consider the restriction to V (u) of f1,..., fn−1. They obviously satisfy all the assumptions of Lemma 4.1, hence by induction form a regular sequence on V (u). Since the local ring of x on V (u) is of the form Ox/(u), we see that u, f1,..., fn−1 is a regular sequence. It then follows from Lemma 4.2 that f1,..., fn−1, u is also a regular sequence. To prove that f1,..., fn−1, fn is a regular sequence, we need only prove that fn is not a zerodivisor of Ox/(f1,..., fn−1). By the assumption on f1,..., fn, in some neighbourhood of x, the equations f1 = · · · = fn = 0 have no solution other than x. Thus the Nullstellensatz shows that (f1,..., fn) ⊃ mk x for some k. In particular uk �
� (f1,..., fn), that is, uk ≡ afn mod(f1,..., fn−1) for some a ∈ Ox. Now if fn were a zerodivisor of Ox/(f1,..., fn−1), it would follow that uk, hence also u, is a zerodivisor of Ox/(f1,..., fn−1). But this contradicts the fact just proved that f1,..., fn−1, u is a regular sequence. Lemma 4.1 is proved, and with it Theorem 4.1. 1.3 Invariance Under Linear Equivalence We come now to the proof of the basic property of intersection numbers, which is the cornerstone of all their applications. Theorem 4.2 Let X be a nonsingular projective variety and D1,..., Dn, D sors such that both D1,..., Dn−1, Dn and D1,..., Dn−1, D tion, and suppose that Dn and D n are linearly equivalent. Then n divin are in general posi- D1 · · · Dn−1Dn = D1 · · · Dn−1D n. (4.9) By the assumption of the theorem Dn − D n = div f, and (4.9) is equivalent to D1 · · · Dn−1 div f = 0, (4.10) when D1,..., Dn−1 and div f are in general position. Representing Di for i = 1,..., n − 1 as a difference of effective divisors, we see that it is enough to prove (4.10) when Di > 0 for i = 1,..., n − 1. We assume this from now on. The proof of Theorem 4.2 uses a notion of intersection number more general than that used so far. Namely, suppose given k ≤ n effective divisors D1,..., Dk on an n-dimensional nonsingular variety X. We say that these are in general position 1 Deﬁnition and Basic Properties k i=1 Supp Di = n − k or if dim satisﬁed, and that k i=1
Supp Di = ∅. Suppose that this property is 239 Supp Di = Cj, k i=1 (4.11) where the Cj are irreducible (n − k)-dimensional varieties. Under these conditions, we can assign a number to each component Cj, called the intersection multiplicity of D1,..., Dk along Cj ; this coincides with the intersection multiplicity at a point if k = n, when each Cj is just a point. The deﬁnition of intersection multiplicity along Cj uses a general notion that we now introduce. Deﬁnition 4.3 A module M over a ring A is of ﬁnite length if it has a ﬁnite chain of A-submodules M = M0 ⊃ M1 ⊃ · · · ⊃ Mn = 0 with Mi = Mi+1, (4.12) such that each quotient Mi/Mi+1 is a simple A-module, that is, does not contain a submodule other than 0 and the module itself. It follows from the Jordan–Hölder theorem that all such chains are made up of the same number n of modules; this common length n is called the length of M, and denoted by (M), or A(M). If A is a ﬁeld, the length of a module becomes simply the dimension of a vector space. If M has ﬁnite length then so do all its submodules and quotient modules. If a module M has a chain (4.12) such that each quotient Mi/Mi+1 has ﬁnite length then also M has ﬁnite length, and (M) = (Mi/Mi+1). The deﬁnition of intersection multiplicity along Cj mimics exactly that of intersection multiplicity at a point. Let C be one of the components Cj in (4.11). We choose a point x ∈ C and local equations fi of the Di in a neighbourhood of x. Then fi ∈ OC (here OC = OX,C is the local ring of X along C, see Section 1.1, Chapter 2), and the ideal a = (f1,..., fk) ⊂ OC does not depend on the choice of the local equations fi or of the point x. Indeed, if g1
,..., gk are other local equations in a neighbourhood of another point of C then the fi and gi are both local equations of Di in a whole open set that intersects C. It follows that the fig are units of OC, and hence (f1,..., fk) = (g1,..., gk). −1 i Lemma 4.3 OC/a is a module of ﬁnite length over OC. Indeed, since C is an irreducible component of the subvariety deﬁned by equations f1 = · · · = fk = 0, there exists an afﬁne open set U ⊂ X intersecting C in which these equations deﬁne C. Then by the Nullstellensatz, (f1,..., fk) ⊃ ar C for some r > 0. Here aC ⊂ k[U ] is the ideal of the afﬁne coordinate ring of U deﬁning C ∩ U. Now set A = k[U ] and p = aC, and consider the local ring Ap and the natural homomorphism ϕ : A → Ap as in Section 1.1, Chapter 2. Then Ap = OC, ϕ(aC) = mC and ϕ((f1,..., fk)) = a. Hence in OC, we have a ⊃ mr C. 240 4 Intersection Numbers The lemma now follows from the following general property of local rings: if a is an ideal of a Noetherian local ring O with maximal ideal m and a ⊃ mr for some r > 0 then O(O/a) < ∞. See Proposition A.16. The lemma is proved. Deﬁnition 4.4 The number OC (OC/a) is called the intersection multiplicity of D1,..., Dk along C, and denoted by (D1 · · · Dk)C. From now on, we consider the case k = n − 1, so that the components Ci of the intersection D1 ∩ · · · ∩ Dn−1 are curves. Write O for the quotient ring Ox/a, where a = (f1,..., fn−1); this is obviously a local ring, with maximal
ideal m the image of the maximal ideal m ⊂ Ox under the quotient homomorphism Ox → O. We ﬁrst need to determine the prime ideals of O. Write pi for the set of functions of Ox that vanish identically on the curve Ci, and pi for its image in O. Obviously O/pi = Ox/pi = OCi,x is the local ring of x on Ci. Lemma 4.4 For a ﬁxed point x ∈ X, suppose that C1,..., Cr are the components of the intersection D1 ∩ · · · ∩ Dn−1 through x. Then p1,..., pr and m are all the prime ideals of O. Proof This is equivalent to saying that p1,..., pr and mx are all the prime ideals of Ox containing a. Let p be a prime ideal with a ⊂ p ⊂ Ox. Consider an afﬁne neighbourhood U of x such that f1,..., fn−1 are regular in U, and set A = k[U ] and P = A ∩ p. Obviously P is a prime ideal. Let V be the subvariety of U deﬁned by P ; because p ⊃ a, clearly V ⊂ C1 ∪ · · · ∪ Cr, and V is irreducible since P is prime. Hence V is either equal to one of the components Ci, and then P = A ∩ pi, or is a point y ∈ U (recall that the Ci are 1-dimensional). In the latter case, if y = x then P, hence also p, contains a function that is nonzero at x. This gives p = Ox in the local ring Ox, but Ox does not count as a prime ideal. Thus the unique remaining possibility is P = A ∩ mx. Since p = P Ox it follows at once that p = pi for i = 1,..., r or p = mx, as asserted in the lemma. The lemma is proved. The ideals pi are obviously minimal prime ideals of O. A local ring in which every prime ideal except for the maximal ideal is minimal is said to be 1-dimensional. Thus O is a 1-dimensional local ring. If f ∈ O is an element of a 1-dimensional local ring which is a non-
zerodivisor then the length (O/(f )) can be expressed in terms of invariants connected with the localisation of O at minimal prime ideals: O/(f ) = O pi Opi (Opi ) × O O/(pi + f O). (4.13) This is a general property of 1-dimensional local rings. The proof is given in Proposition A.17. In our case f = fn, so that O/(f ) = Ox/(f1,..., fn), and therefore the left-hand side is (O/(f )) = (D1 · · · Dn)x. 1 Deﬁnition and Basic Properties For the right-hand side, it is easy to check that Opi (Opi ) = (OCi /a) = (D1 · · · Dn−1)Ci. Finally Opi 241 ∼= Opi /ϕpi (a), so that O/(pi + f O) = (O/pi)/(f ) = OCi,x/(f ), and therefore O(O/(pi + f O)) = (OCi,x/(fn)) = (ρCi (Dn))x, where ρCi (Dn) is the restriction of the divisor Dn to the irreducible curve Ci (see Section 1.2, Chapter 3). Thus (4.13) can be rewritten (D1 · · · Dn)x = r (D1 · · · Dn−1)Ci i=1 × x. ρCi (Dn) (4.14) We now prove that the multiplicity (D)x at a point x of a locally principal divisor D on a curve C is given by the formula (D)x = ν(y)=x ν∗(D) y, (4.15) where ν : Cν → C is the normalisation. Indeed, let f be the local equation of a divisor D in a neighbourhood of a point x ∈ C. Then (4.15) can be rewritten Ox/(f ) = Oy/(f ), ν(y)=x (4.16) where Ox and Oy are the local rings of points x ∈ C and y ∈ Cν. ν(y)=x Write O = Oy. Since O is contained in the �
�eld of fractions of Ox, for every u ∈ O there exists v ∈ Ox such that uv ∈ Ox. According to Lemma of Section 2.1, Chapter 3, O is a ﬁnite Ox -module. Suppose that O = Oxu1 + · · · + Oxur, and for each i, let vi ∈ Ox be such that uivi ∈ Ox ; set v = v1 · · · vr. Then v O ⊂ Ox. It follows in particular that ( O/Ox) ≤ ( O/v O), and by Theorem 3.6, ( O/v O) = ν(y)=x vy(v) < ∞, and hence ( O/Ox) < ∞. From the diagram f O ⊂ O f Ox ⊂ Ox it follows that ( O/(f )) + (f O/f Ox) = ( O/Ox) + (Ox/(f )). Since O has no ∼= f O/f Ox and ( O/Ox) = (f O/f Ox), hence (Ox/(f )) = zerodivisors, O/Ox ( O/(f )). By Theorem 3.6, ( O/(f )) = ν(y)=x (Oy/(f )). This proves (4.16) and (4.15). ν(y)=x vy(f ) = The proof of Theorem 4.2 follows almost at once by combining (4.14) and (4.15). We write the intersection number in the form D1 · · · Dn = (D1 · · · Dn)x. x∈X 242 By (4.14), and by (4.15), 4 Intersection Numbers D1 · · · Dn = r (D1 · · · Dn−1)Cj j =1 × x∈Cj ρCj (Dn) x, ρCj (Dn) = x x∈Cj ν∗ y∈Cν j ρCj (Dn) y. Now if Dn = div f is a principal divisor, with f ∈ k(X), then so are the divisors ν∗(ρCj (Dn)) on the curves Cν j : that is, ν
∗(ρCj (Dn)) = div g, where g = ν∗(ρCj (f )) ∈ k(Cj ), and therefore (div g)y = vy(g). Because X is projective, so are the Cj, and so are their normalisations Cν j by Theorem 2.23. Now by Theovy(g) = deg(div g) = 0, rem 3.5 and Corollary of Section 2.1, Chapter 3, and it follows from this that D1 · · · Dn−1 div f = 0. Theorem 4.2 is proved. y∈Cν j 1.4 The General Deﬁnition of Intersection Number Theorem 4.2, together with Theorem 3.1 on moving the support of a divisor away from a point, enables us to deﬁne an intersection number of any n divisors on an n-dimensional nonsingular projective variety without assuming any restriction such as general position. For this we need two lemmas. Lemma 4.5 For any n divisors D1,..., Dn on an n-dimensional variety X, there exist n divisors D i (linear equivalence) for i = 1,..., n and D 1,..., D n such that Di ∼ D n are in general position. 1,..., D Proof Suppose that we have found divisors D 1,..., k, and either dim(Supp D 1 empty. Let ∩ · · · ∩ Supp D 1,..., D k such that Di ∼ D i for i = k) = n − k or this intersection is Supp D 1 ∩ · · · ∩ Supp D k = C1 ∪ · · · ∪ Cr be its decomposition into irreducible components. We choose a point xj ∈ Cj on each component, and, using the theorem on moving the support of a divisor, ﬁnd a ∼ Dk+1 and xj /∈ Supp D divisor D k+1 for j = 1,..., r. Then a fortiori Supp D k+1 does not contain any of the components Cj, and by the theorem on dimension of intersections k+1 such that D k+1
Supp D dim 1 ∩ · · · ∩ Supp D k+1 = n − k − 1, if this intersection is nonempty. Proceeding in the same way until k = n we get the required system of n divisors. The lemma is proved. 1 Deﬁnition and Basic Properties 243 Lemma 4.6 If D1,..., Dn and D position and Di ∼ D 1,..., D i for i = 1,..., n then n are two n-tuples of divisors in general D1 · · · Dn = D 1 · · · D n. (4.17) Proof If Di = D Let us prove that (4.17) hold if Di = D assertion. i for i = 1,..., n − 1 then this is the assertion of Theorem 4.2. i for i = 1,..., n − k. For k = n we get our We use induction on k. Suppose that the assertion holds for smaller values of k. Since both D1,..., Dn and D 1,..., D n are in general position, both Y = i=n−k+1 Supp Di and Y = i=n−k+1 Supp D i are 1-dimensional. We choose one point on each component of each of Y and Y, and, by the theorem on moving the support of a divisor, ﬁnd a divisor D n−k+1 such n−k+1 does not contain any of these points and D that Supp D ∼ Dn−k+1. Then both D1,..., Dn−k, D n are in general position. Then by Theorem 2 n−k,..., Dn and D n−k+1 n−k,..., D 1,..., D n−k, D D1 · · · Dn = D1 · · · Dn−kD · · · D n and D 1 = D 1 n−k+1 · · · Dn · · · D n−kD n−k+1 · · · D n. (4.18) Now the right-hand sides in (4.18) are equal by induction, since they involve n − k + 1 equal factors. This proves Lem

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