Algebraic surfaces

Algebraic surfaces

Birational maps on surfaces and minimal models Remark 0.1. Most of the material of this section can be found in [Beauville]. The last example is called the elementary transform of ruled surface. It can be found in [Ha, V.5, p.416]

We are going to study birational maps on surfaces and introduce the notion of minimal models in this section.

Recall that by a rational map f : X 99K Y we mean an equivalent classes (U, f ) of morphism f : U → Y . Roughly speaking, a ratio- nal map is a map not everywhere defined. Sometimes the map can be extended to a larger domain. For example, let (U, f ) ∼ (V, g) be equivalent rational maps, i.e. f = gon U ∩ V , then one can define a map on U ∪ V . Indeed, one can extend (U, f ) to U₀ := ∪_{V ∈C}V , where C denotes the equivalent class. The point in X − U₀ is called points of indeterminacy.

Proposition 0.2. Let f : X 99K Y be a rational map to a projective variety Y . Then point of indeterminacy X − U0 has codimension ≥ 2.

In particular, if dimX = 2 then the set of points of indeterminacy is finite.

Proof. Since Y is projective, we may assume that Y = Pⁿ. Let H be a hyperplane in Pⁿ and D = f^∗H. Let Z₀, .., Z_n be the homogeneous coordinates of Pⁿ. Then div(Z_i◦ f ) gives a divisor F_i ∈ |D|.

The point of indeterminacy are exactly the common zero of Zi ◦ f . (Note that Zi◦ f is a section in H⁰(X, O(D)), which is locally regular.

Thus we only need to worry about common zeros)

If W₁ ⊂ X is a codimension 1 subvariety of point of indeterminacy.

Then W₁is the common zero hence W₁ < F_i for all i. Let D₁ = D −W₁ and s ∈ H⁰(X, O(W₁)) a section defining W₁, i.e. div(s) = W₁. One sees that ^Zi_s^◦f ∈ H⁰(X, O(D₁)) which are locally regular.

We consider the map f₁ : X 99K Pⁿby [^Z0_s^◦f, ...,^Zn_s^◦f]. It’s clear that f₁ = f on U₀. (Potentially, f₁ might have eliminated indeterminacy on W₁). Thus f₁ can be extended to a larger defining domain.

By continuing this process, we get f₁, f₂, ... associated to effective divisors D₁ D₂ D₃.... Since effective divisor can have only finitely many non-zero places, this process must terminate. That is, we reach f_n : X → Pⁿ without point of indeterminacy of codimension 1. ¤ Indeed, we can eliminate those finite points of indeterminacy by blowing-ups.

Theorem 0.3 (Elimination of indeterminacy). Let f : X 99K Y be a rational map from a surface to a projective variety Y . Then there exists a morphism p : X⁰ → X which is a composition of blowing-ups, together with a morphism f⁰ : X⁰ → Y such that f⁰ ∼ f ◦ p.

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Proof. Since Y is projective, we may assume that Y = Pⁿ. Let H be a hyperplane in Pⁿ and D = f^∗H. Let Z0, .., Zn be the homogeneous coordinates of Pⁿ. Then div(Z_i◦ f ) gives a divisor F_i ∈ |D|.

The point of indeterminacy are exactly the common zero of Z_i ◦ f . Suppose that x ∈ X is a point of indeterminacy. We consider π : X₁ = Bl_x(X) → X. By composition, one has a map

f₁ : X₁ → X 99K Pⁿ

given by [Z0◦ f ◦ π, ..., Zn◦ f ◦ π]. Note that div(Zi◦ f ◦ π) now gives divisors in |π^∗D|.

Recall that for each i, div(Zi◦f ) passes through x of multiplicity mi. Let m = min_i=0,...,nm_i. It thus follows that div(Z_i◦f ◦π) > mE for each i. Let s ∈ H⁰(X⁰, O_X⁰(E) be the section defining E, i.e. div(s) = E.

We then consider the map (as in the previous Proposition) f₁⁰ : X1 99K Pⁿ,

by [^Z0_s^{◦f ◦π}m , ...,^Zn_s^{◦f ◦π}m ]. One sees that f₁⁰ extends f₁ and f₁⁰ is defined on all but finite point on E.

If there is point of indeterminacy for f₁, we then continue this process to obtain f_k : X_k 99K Pⁿ inductively. It remains to show that this process must stop.

Notice that we may assume that f : X 99K Pⁿ is non-constant and non-degenerate. Thus pick any two general hyperplane Hi, H2 in Pⁿ, one has H₁.H₂.f (X) ≥ 0. Thus f^∗H₁.f^∗H₂ = D² ≥ 0.

Notice that the divisor corresponds to f₁is D₁ := π^∗D−mE, one has D₁² = D² − m² ≥ 0. By applying this observation to all f_i : X_i → Pⁿ. One has

D² D²₁ D²₂... ≥ 0.

Hence it must stop at some Dk, thus one has that fk : Xk → Pⁿhas no point of indeterminacy. Set X⁰ := Xk, f⁰ := fk then we are done. ¤ The following property is crucial in the study of birational map of surfaces.

Proposition 0.4. Let f : X → Y be a birational morphism. If y ∈ Y is a point of indeterminacy of f⁻¹, then f factors through π : Bl_y(Y ) → Y . That is, there is a morphism f⁰ : X → Bl_y(Y ) such that f = π ◦ f⁰. Proof. The proof is pretty long so that we will not include it here.

Please see [Beauville] for the detail. ¤

Corollary 0.5. Let f : X → Y be a birational morphism, then there is π_k : Y_k → Y which is composition of blowing-ups and an isomorphism

² : X → Y_k such that f = π_k◦ ².

Proof. If f is an isomorphism then nothing to prove. It f is not an isomorphism, then there must be a point y ∈ Y such that f⁻¹ is unde- fined at y. One has X → Y₁ := Bl_y(Y ) → Y . One can continue this process unless we have an isomorphism.

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It remains to show that this process must terminate. We need to find an invariant to control the termination. A naive approach is try- ing to count points of indeterminacy at each step. However, this does not behave well because from Y 99K X to Y₁ 99K X, we eliminate the undefining point y but there might have some more point of indeter- minacy on E ⊂ Y₁. Thus we need a more refined invariant.

We consider the rank of Neron-Severi group. Recall that the Neron- Severi group is the algebraic equivalent classes of divisors. It seems difficult to understand what it is. But anyway, it’s an finitely generated abelian group. Moreover,

NS(Blx(X)) = NS(X) ⊕ Z[E].

In particular,

rk(NS(Bl_x(X)) = rk(NS(X)) + 1.

(Remark: If X is defined over C, then NS(X) = im(H¹(X, O_X^∗) → H²(X, Z)) which is of course of finite rank).

Now one has

rk(NS(X))... rk(NS(Y₂)) rk(NS(Y₁)) rk(NS(Y )).

It’s clear that the process of producing Y₁, Y₂... must terminate at Y_k for some k since rk(NS(X)) is finite. Hence one has X ∼= Yk cause otherwise one can produce Yk+1. This completes the proof. ¤ The corollary says that a birational morphism of surfaces is basi- cally composition of blowing-ups and isomorphism. Together with the theorem on elimination of indeterminacy, we have the following:

Corollary 0.6. Let f : X 99K Y be a birational map of surfaces. Then there is a surface Z and morphisms g : Z → X, h : Z → Y such that h ∼ f ◦g , where g, h are composition of blowing-ups and isomorphisms.

Let X be a smooth surface, we can consider Bir(X) to be the bi- rational equivalent class of smooth surfaces which are birational to X. We have seen that any two surfaces in Bir(X) are connected by blowing-ups and isomorphisms.

In what follows, we would like to consider Bir(X)⁰ to be the bira- tional equivalent class modulo isomorphism. Then there is a natural partial ordering on Bir(X)⁰ by [X1] ≥ [X2] if there is a birational mor- phism f : X₁ → X₂, where [X₁] denotes the isomorphic class of X₁. We have seen that [Bl_x(X)] [X] and if [X] ≥ [Y ] then [X] = [Y_k] for some composition of blowing up Y_k → Y .

Our next goal is to show that there exist a minimal element in Bir(X)⁰, which we call it a minimal model of X.

Definition 0.7. A non-singular surface X is minimal if for any mor- phism f : X → Y to a non-singular surface, f is an isomorphism.

(i.e. if [X] ≥ [Y ], then [X] = [Y ] in Bir(X)⁰).

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Theorem 0.8. Let X be a non-singular surface, then there exist a minimal surface X⁰ together with a birational morphism f : X → X⁰. In other words, minimal model exists.

Proof. If X is not minimal, then there is an surface Y and a birational morphism f : X → Y . Since f is composition of isomorphism and blowing-ups. We may assume that there is an X1 and X ∼= Bl(X1).

If X1 is minimal then we are done, otherwise, one has X2 and X1 ∼= Bl(X2) similarly. Thus one has sequence of surfaces

X → X₁ → X₂...

However, rk(NS(X_i+1)) = rk(NS(X_i) − 1. Thus the sequence must

stop at a minimal model. ¤

An convenience way to check minimality for surface is the following:

Theorem 0.9. Let X be a non-singular surface. then X is minimal if and only if X has no (−1)-curves.

Proof. If X has an (−1)-curve, then by CAstelnuovo’s contraction theo- rem, there is a contraction X → X⁰ contracting the (−1)-curve. Hence X is not minimal.

On the other hand, if X is not minimal, then as we have seen above, X ∼= Bl(X1) for some X₁. In particular, there the exceptional divisor

is an (−1)-curve. ¤

However, minimal model is not always unique.

Example 0.10. Let X = C × P¹, where C is a curve of genus ≥ 2. X is a ruled surface by considering π : X → C.

Recall that by a ruled surface, we mean a surface X together with a morphism π : X → B to a curve B such that each fiber F_b := π⁻¹(b) ∼= P¹.

Fix now a point x ∈ X lying over b ∈ C. We consider Z = Bl_x(X).

And there is a composition map π_Z : Z → C. Now over b ∈ C, π_Z⁻¹(b) = ˜F_b+E. Easy computation show that ˜F_b is a (−1)-curve on Z.

One can contract ˜F_b and obtained a surface Y . There is a π_Y : Y → C.

But one can prove that Y 6∼= C × P¹ = X.

However, both X and Y are minimal model of Z. Hence minimal model is not unique.