COMPLEX ALGEBRAIC SURFACES CLASS 7

RAVI VAKIL

Contents

1. How basic aspects of surfaces change under blow-up 1 2. Rational maps of surfaces, linear systems, and elimination of indeterminacy 3 3. The universal property of blowing up 4 3.1. Applications of the universal property of blowing up 4

Recap of last time. Last time we began discussing blow-ups:

Given p ∈ S, there is a surface S′ = Bl_{p}S and a morphism π : S′ → S, unique up to isomorphism, such that (i) the restriction of π to π^{−1}(S − {p}) is an isomorphism onto S − {p}, and (ii) π^{−1}(p) is isomorphic to ℙ^{1}. π^{−1}(p) is called the exceptional divisor p, and is called the exceptional divisor.

A key example, and indeed the analytic-, formal-, or etale-local situation, was given by blowing up S = 𝔸^{2} at the origin, which I’ll describe again soon when it comes up in a proof.

For the definition, complex analytically, you can take the same construction. Then you need to think a little bit about uniqueness. There is a more intrinsic definition that works algebraically, let I be the ideal sheaf of the point. Then S′ = Proj ⊕_{d≥0}I^{d}.

1. How basic aspects of surfaces change under blow-up

Definition. If C is a curve on S, define the strict transform C^{strict} of C to the the closure of the pullback on S − p, i.e. \overline{π|_{S′−E}^{∗}(C ∩ S − p)}. The proper transform C^{proper} is given by the pullback of the defining equation, so for example π^{∗}O_{S}(C) = O_{S′}(C′).

Lemma. If C has multiplicity m at p, then C^{proper} = C^{strict} + mE, i.e. π^{∗}C ≅ C^{strict} + mE.

Proof. The multiplicity of C being m means that in local coordinates, the defining equation has terms of degree m, but not lower. (Better: the defining equation lies in 𝔪^{m} but not 𝔪^{m+1}.) Analytically, this means that the leading term in x and y has degree m.

Date: Wednesday, October 23.

We do this by local calculation which will be useful in general. (Draw picture.) U_{0} = {((x_{0}, y_{0}), [1; v]) : y_{0} = x_{0}v} = Spec k[x_{0}, y_{0}, v]/y_{0} = x_{0}v = Spec k[x_{0}, v]. The exceptional divisor E is given by y_{0} = 0 (after morphism).

U_{1} = {((x_{1}, y_{1}), [u; 1]) : x_{1} = y_{1}u} = Spec k[x_{1}, y_{1}, u]/x_{1} = y_{1}u = Spec k[y_{1}, u]. The exceptional divisor E is given by y_{1} = 0.

Map down to (x, y) = Spec k[x, y]. (x_{0}, y_{0}, v) ↦ (x_{0}, y_{0}), (x_{1}, y_{1}, v) ↦ (x_{1}, y_{1}).

Given a function f(x, y) = 0. Pull it back to U_{0}: f(x, y) = f_{m}(x, y)+higher = f_{m}(x_{0}, x_{0}v)+ higher + · · ·.

Exercise. To see if you understood that, do the same calculation on patch 2.

Theorem. Suppose π : S′ → S is a blow up of S at p, with exceptional curve E ⊂ S′. Let D and D′ be divisors on S. Then π^{∗}D · π^{∗}D′ = D · D′, E · π^{∗}D = 0, E^{2} = −1.

Remark. A curve on a smooth surface that is isomorphic to ℙ^{1} and has self-intersection −1 is called a (−1)-curve.

Proof. The first we did yesterday. The second: by Serre’s moving lemma, we can move D away from p, then pull back. For the third: choose a curve C passing through p with multiplicity 1. (How to do this: hyperplane section of S.) Then C^{strict} · E = 1. Also C^{proper} · E = 0. as C^{strict} + E = C^{proper}, we’re done.

Theorem. (a) There is an isomorphism Pic S ⊕ ℤ →^{∼} Pic S′ defined by (D, n) ↦ π^{∗}D + nE.

(b) The same with Pic replaced by NS.

Proof. The arguments are the same for both parts, so I’ll do (a). It is surjective: the divisors upstairs are either E or strict transforms (which are proper transforms plus E’s). It is injective: if π^{∗}D + nE = 0, then intersect with E to see that n = 0; then apply π_{∗} to see that D = 0.

Theorem. K_{S′} = π^{∗}K_{S} + E.

Proof. Clearly K_{S′} = π^{∗}K_{S} + mE for some m. By the adjunction formula for E, K_{E} = K_{S′}(E)|_{E}. Taking degrees:

−2 = (π^{∗}K_{S} + mE + E) · E = −m − 1.

Exercise/Remark. If you want practice with the canonical bundle in local coordinates, take a meromorphic section of K_{S} that has neither zero nor pole at p (possible by Serre’s moving lemma), write it as f(x, y)dx ∧ dy, and pull it back to the open set U_{1} to see that you get f(x_{0}, x_{0}v)dx_{0} ∧ d(x_{0}v) = f(x_{0}, x_{0}v)x_{0}dx_{0} ∧ dv.

2. Rational maps of surfaces, linear systems, and elimination of indeterminacy

A rational map S −−→ X, where X is a variety, means a morphism from an dense open set of S. Recall that a rational map from a curve C to a projective variety can always be extended to a morphism. Similarly, a rational map from a surface S to a projective variety can be extended over most points; the set of indeterminacy is a finite set of points. More precisely, given a map π : S −−→ ℙ^{n}. This is given by n + 1 sections of some line bundle. It makes sense except where the sections are all zero. This will be in codimension 2.

Let F be this finite set. We’ll denote \overline{π(S − F)} the image of S, and denote it π(S). (I’m not sure we need to take the closure.) If C is a curve on S, then we’ll denote \overline{π(C − F)} the image of C, and denote it π(C). Here we definitely need to take the closure.

Now suppose you have a divisor D on S. Given a subspace V of dimension n of H^{0}(S, O(D)), we might hope to get a map to projective space ℙV^{∗}. (This is called a linear system of dimension n; I should have introduced this notation earlier.) If it is base point free, we do.

If it has base points, the locus could have components of dimension 1. Such a component is called a fixed component of the linear system V. The fixed part of V is the biggest divisor contained in every element of V. So if this fixed part is F, then D − F has no fixed components.

(I’m not happy with how I explained the previous paragraphs in class. I hope this is clearer.)

Lemma. If the linear systems has no fixed part, then it has only a finite number of fixed points.

Proof. Take two general sections, and look at their two zero-sets. Where do they intersect? At a bunch of points. Hence we get at most D^{2}?

We’ve basically shown that there is a bijection between:

(i) { rational maps π : S −−→ ℙ^{n} such that π(S) is contained in no hyperplane }

(ii) { linear systems on S without fixed part and of dimension n }

(Explain the correspondence.)

Theorem (Elimination of indeterminacy). Let π : S −−→ X be a rational map from a surface to a projective variety. Then there exists a surface S′, a morphism η : S′ → S which is the composite of a finite number of blow-ups, and a morphism f : S′ → X such that the diagram

S′
↙^{η}  ↘^{f}
S  −−→^{π}  X

is commutative.

Proof. Idea: blow up fixed points, show that D^{2} decreases.

We immediately reduce to the case where X is ℙ^{m}, and π(S) isn’t contained in any hyperplane of ℙ^{m}. Then ϕ corresponds to a linear system V ⊂ |D| of dimension n on S, with no fixed component. If V has no base point, then we’re done.

Otherwise, we blow up a base point x, and consider S_{1} → S at x (and hence a rational map S_{1} −−→ S). The exceptional curve is now in the fixed part of the linear system, with some multiplicity k ≥ 1. So we subtract kE to get rid of the fixed part, i.e. get a new linear system V_{1} ⊂ |π^{∗}D−kE|, to get the same rational map ϕ_{1} : S_{1} −−→ S, given by D_{1} = D−kE. If this is a morphism, we win, otherwise we keep going.

At some point, this process must stop (and hence we win in the long run). We prove this is the case when D^{2} = i, by induction on i. Base case, i = 0: the number of fixed points is bounded by D^{2} = 0, so there aren’t any. Inductive step: Now i > 0. Then we blow-up once, and we get a new surface with divisor class. On this surface, D_{1}^{2} = (D − kE)(D − kE) = D^{2} − k^{2} < D^{2}. So by the inductive hypothesis, the process will terminate on this new surface, completing the induction.

3. The universal property of blowing up

Theorem (Universal property of blowing up). Let f : X → S be a birational morphism of surfaces, and suppose that the rational map f^{−1} is undefined at a point p of S. Then f factorizes as

f : X →^{g} S̃ := Bl_{p} S →^{π} S

where g is a birational morphism and π is the blow-up at p.

Proof: next day.

3.1. Applications of the universal property of blowing up. Two theorems.

Theorem (all birational morphisms factor into blow-ups). Let f : S → S_{0} be a birational morphism of surfaces. Then there is a sequence of blow-ups π_{k} : S_{k} → S_{k−1} (k − 1, …, n) and an isomorphism u : S →^{∼} S_{n} such that f = π_{1} ∘ ⋯ ∘ π_{n} ∘ u.

Proof. If f is an isomorphism, we’re done. Otherwise, there is a point p of S_{0} such that f^{−1} is undefined At p, and we can factor through S → S_{1} = Bl_{p} S_{0}. We can repeat this.

If n(f_{k}) is the number of contracted curves of n(f_{k}) < n(f_{k−1}): if E is the exceptional divisor of π_{k} : S_{k} → S_{k−1}, then the preimage of E in S contains a curve which is contracted by f_{k−1} but not f_{k}. As the number of contracted curves can’t be negative, the process must terminate.

Theorem (all birational maps can be factored into blow-ups). Let ϕ : S −−→ S′ be a birational map of surfaces. Then there is a surface S′′ and a commutative diagram

S′′
↙^{f}  ↘^{g}
S  −−→^{ϕ}  S′

where the morphisms f and g are composites of blow-ups.

Proof. By the theorem of elimination of indeterminacy, we can find such a diagram such that f is a composition of blow-ups. By the Theorem above, g must then be a composition of blow-ups too.

We’ve now proved some powerful stuff, so let’s take a step back and see what we now know, and how it relates to classification.

Two surfaces are birational iff they can be be related by sequences of blow-ups. We’ll be interested in birational classification, but biregular classification is very close.

If f : S → S′ is birational which is the composition of n blow-ups, then NS(S) ≅ NS(S′) ⊕ ℤ^{n}, so n is independent of the choice of blow-ups. Exercise: Use this to show that every birational morphism from S to itself is an isomorphism.

Fact. In a blow-up, H^{i} of the structure sheaf is preserved, i.e. if π : S′ → S is a blow-up, then π^{∗} : H^{i}(O_{S}) → H^{i}(O_{S′}) is an isomorphism.

The algebraic way of proving this fact comes from the Leray spectral sequence, and the fact that π_{∗}O_{S′} = O_{S} and R^{i}π_{∗}O_{S′} = 0 for i > 0. This in turn requires some infinitesimal analysis, in the form of “formal function theorems”. I suspect that there should be a relatively straightforward analytic proof.

In particular, by these numbers are birational invariants.

So look at what this means for the Hodge diamond. When you blow up, you add 1 to the central entry (the rank of the Neron-Severi group). Everything else is constant.

Next day: More consequences of these powerful theorems. Proof of the universal property of blowing up. Castelnuovo’s criterion for blowing down curves.