INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 25

RAVI VAKIL

Contents

1. The genus of a nonsingular projective curve 1 2. The Riemann-Roch Theorem with Applications but No Proof 2 2.1. A criterion for closed immersions 3 3. Recap of course 6

PS10 back today; PS11 due today. PS12 due Monday December 13.

1. The genus of a nonsingular projective curve

The definition I’m going to give you isn’t the one people would typically start with. I prefer to introduce this one here, because it is more easily computable.

Definition. The tentative genus of a nonsingular projective curve C is given by deg Ω_{C}^{1} = 2g − 2.

Fact (from Riemann-Roch, later). g is always a nonnegative integer, i.e. deg K = −2, 0, 2, . . ..

Complex picture: Riemann-surface with g “holes”.

Examples. Hence ℙ^{1} has genus 0, smooth plane cubics have genus 1, etc.

Exercise: Hyperelliptic curves. Suppose f(x_{0}, x_{1}) is a polynomial of homogeneous degree n where n is even. Let C_{0} be the affine plane curve given by y^{2} = f(1, x_{1}), with the generically 2-to-1 cover C_{0} → U_{0}. Let C_{1} be the affine plane curve given by z^{2} = f(x_{0}, 1), with the generically 2-to-1 cover C_{1} → U_{1}. Check that C_{0} and C_{1} are nonsingular. Show that you can glue together C_{0} and C_{1} (and the double covers) so as to give a double cover C → ℙ^{1}. (For computational convenience, you may assume that neither [0; 1] nor [1; 0] are zeros of f.) What goes wrong if n is odd? Show that the tentative genus of C is n/2 − 1. (This is a special case of the Riemann-Hurwitz formula.) This provides examples of curves of any genus.

Date: Thursday, December 9, 1999.

Exercise. Suppose C is a nonsingular degree d projective plane curve. Show that the tentative genus of C is \binom{d−1}{2}.

2. The Riemann-Roch Theorem with Applications but No Proof

Definition. If X is a topological space, and S is a sheaf on X, define H^{0}(X, S) to be the set of global sections of S over X (what we’ve called S(X)). If X is a variety, and S is an O_{X}-module, then H^{0}(X, S) has the structure of a \overline{k}-vector space. In this situation, define h^{0}(X, S) to be the dimension of H^{0}(X, S) as a \overline{k}-vector space.

From here on in, C is a nonsingular projective curve, and we’ll deal with the case when S is an invertible sheaf. In this case, we have shown that h^{0} is finite.

The Celebrated Riemann-Roch Formula. L a line bundle of degree d on a nonsingular projective curve C of genus g. Then h^{0}(C, L) − h^{0}(C, Ω^{1} ⊗ L^{∨}) = d − g + 1.

Proof. Omitted!

I think you know enough to understand a proof, or at least a proof under mild assumptions. If there are 2 or 3 of you interested, I could try to come up with a proof that you’d understand in half an hour, and present it during IAP.

Example. Let’s check this out for ℙ^{1}. (Make a table of degree, h^{0}(C, L), h^{0}(C, Ω^{1} ⊗ L^{∨}), d − g + 1.)

Here come lots of appliations.

Corollary. Plug in L = O. Then we see that h^{0}(C, Ω^{1}) = g. Hence the tentative genus is always a nonnegative integer.

Definition. Define the genus of a nonsingular projective curve C to be h^{0}(C, Ω_{C}^{1}).

Corollary. If d > 2g − 2 (i.e. the degree of L is greater than the sheaf of differentials), then h^{0}(C, Ω^{1} ⊗ L^{∨}) = 0. Hence h^{0}(C, L) = d − g + 1.

Reason: the invertible sheaf Ω^{1} ⊗ L^{∨} has negative degree.

Corollary. Given p ∈ C. Then there is a rational function with a pole only at p.

Proof. Consider the divisor (2g +1)p. Then L := O((2g +1)p) is a degree 2g line bundle, and it has h^{0}(C, L) = d−g +1 = 2g +1−g +1 = g +2 > 1. Hence there are two linearly independent sections of L. But the global sections of L corresponded to rational functions whose valuation at every point distinct from p was nonnegative, and whose valuation at p was at most −(2g + 1). One such function is the constant function; thus there is a function that is nonconstant, which must therefore have a pole somewhere; the pole must be at p. (We also see that the order of pole can be made to be at most 2g + 1.)

2.1. A criterion for closed immersions. Given a nonsingular projective curve C, and invertible sheaf L. Let {s_{0}, …, s_{n}} be a basis for the (finite-dimensional) vector space H^{0}(C, L) of global sections. If, for each p ∈ C, there is a section s of L such that s doesn’t vanish at p, then this basis gives a morphism

(s_{0}, …, s_{n}) : C ↦ ℙ^{n}.

This morphism is denoted |L|, and is called the complete linear system associated with L.

Remark. The sections of L ⊗ O_{C}(−p) (often written L(−p)) over U can be identified with the sections of L vanishing at p. (You’ve seen this before if L is O_{C}; the proof is the same.) Hence there is a natural inclusion

H^{0}(C, L(−p)) ↪ H^{0}(C, L).

The cokernel is dimension 0 or 1; it is 1 if there is a global section of L that does not vanish at p. Hence we’ve proved:

Proposition. h^{0}(C, L) − h^{0}(C, L ⊗ O_{C}(−p)) ≤ 1, and if equality holds then then there is a section of L that does not vanish at p.

Proposition. Suppose p and q are distinct points of C. If h^{0}(C, L) − h^{0}(C, L ⊗ O_{C}(−p − q)) = 2, then p and q are mapped to distinct points of ℙ^{n} via |L|. In other words, |L| is injective.

Proof. (You can check that h^{0}(C, L) − h^{0}(C, L(−p − q)) ≤ 2, and that the hypotheses of this proposition imply the hypotheses of the previous one, so that L really does induce a morphism into projective space.)

It suffices to show that there is a section of L that vanishes at p but doesn’t vanish at q. (Explain.)

When the degree of L is big, the hypotheses of these propositions are automatically satisfied.

Proposition. If deg L > 2g − 1, then for each p ∈ C, there is a section s of L such that s doesn’t vanish at p. If deg L > 2g, then the morphism |L| is injective.

Proof. In this range, h^{0}(C, L) = d − g + 1, h^{0}(C, L(−p)) = (d − 1) − g + 1, and (in the second case) h^{0}(C, L(−p − q)) = (d − 1) − g + 1.

Under the conditions of the second proposition, |L| is an injection of sets. But it still might get knotted up at some point (explain). It would be great to know criteria for when |L| gives a closed immersion. Here’s one:

Proposition. Suppose for any pair of points p and q of C (not necessarily distinct), h^{0}(C, L) − h^{0}(C, L ⊗ O_{C}(−p − q)) = 2. Then |L| is a closed immersion.

Proof. Omitted. Shafarevich p. 217; Hartshorne.

Remark. By the same argument as above, if deg L > 2g, then the hypotheses are satisfied.

This is powerful. Here are applications.

Proposition. A projective nonsingular curve C of genus 0 must be isomorphic to ℙ^{1}.

Proof. Choose any point p ∈ C. Then L := O(p) has degree 1 > 2g, so |L| is a closed immersion. But L has 2 sections by Riemann-Roch: h^{0}(C, L) = deg L − g + 1 = 2. Thus |L| describes a closed immersion of C into ℙ^{1}, so C must be isomorphic to ℙ^{1}.

Thus there is only one projective nonsingular genus 0 curve.

Let’s try a similar trick in genus 1. Let C be a genus 1 (projective nonsingular) curve, and let L be any degree 3 invertible sheaf (e.g. O(3p)). Then by Riemann-Roch, h^{0}(C, L) = 3 − 1 + 1 = 3.

Then by the Proposition, |L| gives a closed immersion of C into ℙ^{2}, and it must be as a cubic curve. (In fact the degree of the image is precisely the degree of the line bundle. We haven’t analyzed this carefully yet, so I’ll resort to another argument: from an exercise in PS12, you can calculate that the genus of a degree d nonsingular plane curve is \binom{d−1}{2}. Hence as the image of |L| has genus 1, we know d = 3.)

Hence we have proved:

Proposition. Every genus 1 curve can be written as a cubic plane curve.

Thus we’ve “seen” all genus 1 curves. By special request, we proved that:

Proposition. There is a natural bijection between the degree 1 line bundles on a genus 1 curve C and the points of the curve.

Proof. The link is as follows. If p is a point on the curve, then O_{C}(p) is a degree 1 invertible sheaf. If L is a degree 1 invertible sheaf, then by Riemann-Roch, h^{0}(C, L) = 1, so there is (up to multiple) precisely one section of L. It’s divisor is necessarily a point. These two constructions clearly commute.

In the problem set, you’ll see that all genus 2 curves can be written as double covers of ℙ^{1} branched at six points. So now we’ve “seen” all genus 2 curves.

In the problem set, you’ll see that all genus 3 curves can be written either as double covers of ℙ^{1} branched at eight points, or as nonsingular plane quartics. So now we’ve “seen” all genus 3 curves.

Proposition. A projective nonsingular curve minus a finite number of points is affine.

Proof. Consider the invertible sheaf O(kp) where k is huge. (Exercise: how big does k have to be for this to work?) Then this gives a morphism into projective space

|O(kp)| : C → ℙ^{n}

expressing C as a closed subvariety. Now there is a section s_{0} of O_{C}(kp) that has only one zero, of order k, at the point p. Take a basis for the sections of O_{C}(kp): s_{0}, …, s_{n}. So we have a map

(s_{0}, …, s_{n}) : C → ℙ^{n}
q ↦ (s_{0}(q); …; s_{n}(q)).

Now where does x_{0} = 0 intersect C? Answer: where s_{0} = 0, i.e. only at the point p. (Geometry: the curve meets the hyperplane x_{0} = 0 with multiplicity k there.) Hence the affine open U_{0} of C is C \ {p}, so C \ {p} is affine!

Corollary. Any nonsingular curve that is not projective is affine.

Let’s calculate more things.

Suppose C is a genus 1 curve. Let L be any degree 4 invertible sheaf on C, e.g. 4p for some point p. Then |L| expresses C as a closed subvariety of ℙ^{3} (as L has 4 sections, and the degree satisfies the criterion for closed immersions above). I claim that it is the intersection of 2 quadrics. Here is a pretty good sketch of why.

Let π be the inclusion. Let w, …, z be the coordinates on ℙ^{3}. First, note that the image isn’t contained in any hyperplane.

Next, compute the number of sections of π^{∗}O(1), which has degree 4: by Riemann-Roch, it has 4 sections, which are precisely the 4 sections we used to map it to projective space. They correspond to the linear functions w, x, y, z.

Compute the number of sections of π^{∗}O(2), which has degree 8: by Riemann-Roch, it has 8 sections. But here are 10 sections: the sections w^{2}, …, yz, z^{2}. Hence there is a two-dimensional vector space of polynomials that pullback to zero. Let p(w, x, y, z) and q(w, x, y, z) be a general basis of this two-dimensional vector space. Then the image of C lies on p = 0 and q = 0 (explain).

We know that p = 0 and q = 0 must intersect in a curve. Here’s why: if their intersection included a surface S, the surface must be of degree 2 or degree 1. If S were degree 2, then it must be the surface p = 0 and q = 0; but then p is a scalar multiple of q, contradicting the fact that p and q formed a basis. If S were degree 1, then p and q both consist of 2 planes (explain, and explain the contradiction).

Hence p = 0 and q = 0 intersect in a curve C′ (including C). By the exercises on the Hilbert polynomial, the curve C′ has degree 4, which means that it meets a general hyperplane at 4 points.

But a general hyperplane corresponds to the zeros of a general section of L. Any nonzero section of L has 4 zeros, so C is all of C′. Hence C is the intersection of 2 quadrics in ℙ^{3}.

(Conversely, if you take 2 general quadrics in ℙ^{3}, it turns out that you get an elliptic curve.)

Remark. You can use this method to understand curves of genus 4 and 5.

For a genus 4 curve C, the sheaf of differentials has degree 6, and has 4 sections. If C is not hyperlliptic, then |Ω^{1}| expresses C as a closed subvariety of ℙ^{3}, (after an application of the crition above). The methods we just used show that C is a complete intersection of a quadric and a cubic. Hence this “describes all genus 4 curves”: they are either hyperelliptic, or the intersection of a quadric and a cubic. (There is a uniform way of describing them, that doesn’t involve these cases.)

For a genus 5 curve C, something similar happens. C is either hyperelliptic, or it sits in ℙ^{4} as the intersection of 3 quadrics. Not quite! Could be trigonal, in which case the intersection of the 3 quadrics is, not surprisingly, a rational scroll. Johan showed me a paper of Rutger Noot.

3. Recap of course

Generalities: Algebraic sets. Zariski topology. Nullstellensatz. Hilbert basis theorem. Sheaves and stalks; structure sheaf. Prevarieties and morphisms of prevarieties. Function fields and birational maps. Projective prevarieties. Schemes. Products. Dimension. Nonsingularity.

Curves: Valuation rings. Completitions. Integral closure, Dedekind domains, DVRs. Extending rational maps of nonsingular curves to projective varieties. Finitely generated fields over \overline{k} of tr. deg. 1 correspond to nonsingular projective curves.

Line bundles and invertible sheaves. O(m). Picard group. Class group. Degree. Riemann-Roch.