COMPLEX ALGEBRAIC SURFACES CLASS 4

RAVI VAKIL

Contents

1. Serre duality and Riemann-Roch; back to curves 1 2. Applications of Riemann-Roch 2 2.1. Classification of genus 2 curves 3 2.2. A numerical criterion for very ampleness. 3 3. Normal bundles; the adjunction formula 3 3.1. Applications of the adjunction formula 4 3.2. Classification of curves of genus 3 5

Recap of last time. Extended example of ℙ^{n}. O_{ℙ^{n}}(d) ↔ degree d homogeneous things.

Maps to projective space corresponded to vector spaces of sections of an invertible sheaf L that are basepoint free (no common zero). Hyperplane sections correspond to H^{0}(X, L).

For example, if the sections of L have no common zero, then we can map to some projective space by the vector space of all sections. Then we say that the invertible sheaf is L is basepoint free. (I didn’t give this definition last time.)

Definition. The corresponding map to projective space is called a linear system. (I’m not sure if I’ll use this terminology, but I want to play it safe.)

|L| : X → ℙ^{n} = ℙH^{0}(X, L)^{∗}.

Definition. An invertible sheaf L is very ample if the global sections of L gives a closed immersion into projective space.

Fact. equivalent to: “separates points and tangent vectors”.

Definition. An invertible sheaf is ample if some power of it is very ample.

Note: A very ample sheaf on a curve has positive degree. Hence an ample sheaf on a curve has positive degree. We’ll see later today that this is an “if and only if”.

Date: Friday, October 11.

Fact (Serre vanishing). Suppose M is any coherent sheaf e.g. an invertible sheaf, or more generally a locally free sheaf (essentially, a vector bundle), and L is ample. Then for n >> 0, H^{i}(X, M ⊗ L^{n}) = 0 for i > 0.

1. Serre duality and Riemann-Roch; back to curves

Fact: Serre duality. If X is proper nonsingular and dimension n, then for 0 ≤ i ≤ n,

H^{i}(X, L) ⊗ H^{n−i}(X, K ⊗ L^{∨}) → H^{n}(X, K) ∼ ℂ

is a perfect pairing.

(True for vector bundles. More general formulation for arbitrary coherent sheaves.) Thus h^{i}(X, L) = h^{n−i}(X, K ⊗ L^{∗}) and χ(C, L) = (−1)^{n}χ(C, K ⊗ L^{∨}).

In particular, we have Serre duality for curves. For any invertible sheaf L, the map

H^{0}(C, L) ⊗ H^{1}(C, K ⊗ L^{∨}) → H^{1}(C, K) ∼ ℂ

is a perfect pairing.

Hence two possible definitions of genus are the same: h^{0}(C, K) and h^{1}(C, O_{C}). Fact: Riemann-Roch Theorem.

h^{0}(C, L) − h^{1}(C, L) = deg(L) − g + 1.

(I.e. χ(C, L) = deg −g + 1 — rememeber that the cohomology of a coherent sheaf vanishes above the dimension of the variety.)

Generalizations: Hirzebruch-Riemann-Roch, to Grothendieck-Riemann-Roch. Hirzebruch-Riemann-Roch, which we’ll be using, is a consequence of the Atiyah-Singer index theorem, which Rafe Mazzeo spoke about in the colloquium yesterday.

Proof of Riemann-Roch: (i) algebraic, (ii) complex-analytic, (iii) Atiyah-Singer.

Using Serre duality:

h^{0}(C, L) − h^{0}(C, K ⊗ L^{∨}) = deg(L) − g + 1.

Corollary. deg K = 2g − 2. (Do it.)

So we have another definition of g.

2. Applications of Riemann-Roch

Theorem. If deg L ≥ 2g − 1, then h^{1}(C, L) = 0. Hence h^{0}(C, L) = deg L − g + 1.

Proof: Serre duality, and the fact that invertible sheaves of negative degree have no sections. Then Riemann-Roch.

Theorem (numerical criterion for basepoint freeness). If deg L ≥ 2g, then L is basepoint free. (Remind them of definition.)

Proof: H^{0}(C, L(−p)) is the vector space of sections with a zero at p. Want to show that H^{0}(C, L) − H^{0}(C, L(−p)) > 0. Do it using previous result: difference is 1!

Example. If C is genus 1 and L is degree 2, then we get a map to ℙ^{1}. This shows that every genus 1 curve is a double cover of ℙ^{1}. (Explain “hyperplane section” in this case.)

2.1. Classification of genus 2 curves. Theorem. Any genus 2 curve has a unique double cover of ℙ^{1} branched over 6 points. (The 6 points are unique up to automorphisms of ℙ^{1}.) Any double cover of ℙ^{1} branched over 6 points comes from a genus 2 curve. Hence genus 2 curves are classified by the space of 6 distinct points of ℙ^{1}, modulo automorphisms of ℙ^{1}. In particular, there is a dimension 6 − dim Aut ℙ^{1} = 3 moduli space.

Proof. First, every genus 2 curve C has a degree 2 map to ℙ^{1} via K (basepoint freeness). The numerical criterion for basepoint freeness doesn’t apply, unfortunately. Let’s check that K has 2 sections: h^{0}(K) − h^{0}(O) = deg(K) − 2 + 1 = 1. Next, let’s check that K is basepoint free. We want K(−p) has only 1 section for all p. If K(−p) had two sections, then we’d have a degree 1 map to ℙ^{1} from C (explain), contradiction.

Next we’ll see that the only degree 2 maps to ℙ^{1} arise from the canonical bundle. Suppose L is degree 2, and has 2 sections. Then by RR, K ⊗ L^{∗} has 1 section, and is degree 0. But the only degree 0 sheaf with a section is the trivial sheaf.

Finally, any double cover of ℙ^{1} branched over 6 points is genus 2. This follows from the Riemann-Hurwitz formula.

Using naive geometry: 2χ(ℙ^{1}) − 6 = χ(C), i.e. −2 = 2 − 2g, so g = 2..

Fact. Riemann-Hurwitz formula: K_{C} = π^{∗}K_{ℙ^{1}}+ ramification divisor. Taking degrees: 2g − 2 = 2(−2) + 6.

2.2. A numerical criterion for very ampleness. Theorem. If deg L ≥ 2g + 1, then L is very ample. (Remind them of definition.)

Proof. I’ll show you that it separates points. Suppose p ≠ q. H^{0}(C, L) − H^{0}(C, L(−p − q)) = 2. Hence there is a section vanishing at p and not at q, and vice versa.

Same idea works for separating tangent vectors. h^{0}(C, L) − h^{0}(C, L(−2p)) = 2. Hence there is a section vanishing at p but not to order 2.

Example. C genus 1, L has degree 3. Then C ↪ ℙ^{2}, degree 3. Hence every elliptic curve can be described by a plane cubic. (We’ll soon see that any plane cubic is a genus 1 curve.)

Corollary. L on C is ample iff it has positive degree.

3. Normal bundles; the adjunction formula

There is one more natural place to find line bundles: the normal bundle to a submanifold of dimension 1 less. I’ll now discuss normal bundles algebraically, and as an application prove the adjunction formula.

Suppose D is a nonsingular divisor (codimension 1) on nonsingular X. Then we can understand the canonical sheaf of D in terms of the canonical sheaf of X.

The adjunction formula. K_{D} = K_{X}(D)|_{D}. This is sometimes (sloppily) written as K_{D} = K_{D}(D). Remember this!

(Remind them what K_{X}(D) means.)

This actually holds in much more generality, e.g. D can be arbitrarily singular, and X need not be smooth.

Here is an informal description of why this is true.

• Motivation: tangent space in differential geometry to a point p in a manifold W is the space of curves, modulo some equivalence relation.

• The cotangent space is the dual of this space, and can be interpreted as the space of functions vanishing at p modulo an equivalence relation: the functions vanishing to order 2 at p. In the space O of functions near p, the first is the ideal I_{p}, the second is the ideal I_{p}^{2}. Thus the cotangent space is I_{p}/I_{p}^{2}.

• Aside: in this case, I_{p} is a maximal ideal 𝔪_{p}, as O/I_{p} is a field. So the cotangent space is 𝔪_{p}/𝔪_{p}^{2}. That’s why the Zariski tangent spaced is defined as 𝔪_{p}/𝔪_{p}^{2}. This is a purely algebraic definition. It works for any algebraic X, not even defined over a field, not even non-singular.

• Next consider Y to be a nonsingular subvariety of X, of codimension c. Then there is a conormal bundle of Y in X (the dual of the normal bundle. It has rank d. The conormal sheaf is the sheaf of sections of Y. It is locally free of rank d. Definition (motivated by previous discussion). In sheaf language: the conormal bundle is I/I^{2}. This is a priori a sheaf on all of X, but in fact it lives on Y (“is supported on Y ”).

• Now suppose further that Y is a divisor, so I’ll now call it D. Then I = O_{X}(D). We’re modding out this sheaf by functions vanishing on D; this is the same as restricting to D. Hence I/I^{2} ∼ O(D)|_{D}.

Proof of the adjunction formula. We have an exact sequence

0 → T_{D} → T_{X}|_{D} → N_{D} → 0.

Dualize to get:

0 → N_{D}^{∗} → Ω_{X}|_{D} → Ω_{D} → 0.

Take top wedge powers to get K_{X}|_{D} ∼ K_{D} ⊗ N_{D}^{∗} from which K_{D} ∼ K_{X}|_{D} ⊗ N_{D} = K_{X}|_{D} ⊗ O_{X}(D)|_{D} = K_{X}(D)|_{D}.

3.1. Applications of the adjunction formula. 1) Cubics in ℙ^{2} have trivial canonical bundle. Hence all genus 1 curves have canonical sheaf that is not degree 0, but also trivial.

2) Quartics in ℙ^{3} also have trivial canonical bundle. K3 surfaces.

3) What’s the genus of a smooth degree d curve in ℙ^{3}? Answer: (d − 1)(d − 2)/2.

4) Smooth quartics in ℙ^{2} are embedded by their canonical sheaf. More on this in a second.

5) Smooth complete interesction of surfaces of degree 2 and 3 in ℙ^{3}: also embedded by their canonical sheaf. 3.2. Classification of curves of genus 3. I’ll discuss at greater length next day. Theorem. Every smooth curve of genus 3 is of (precisely) one of the two following forms.

(i) A smooth quartic curve in ℙ^{2} (in a unique way, up to automorphisms of ℙ^{2}).

(ii) A double cover of ℙ^{1} branched over 8 points (in a unique way, up to automorphisms of ℙ^{2}).