The minimal model program for surfaces

5.6 The minimal model program for surfaces

LetX be a smooth projective surface. It follows from Castelnuovo’s criterion (Theorem 5.12) that by contracting exceptional curves onX one arrives even-tually (the process must stop because the Picard number decreases by 1 at each step by Exercise 4.8.1)) at a surface X0 with no exceptional curves. Such a surface is called aminimal surface. According to the cone theorem (§5.4),

• eitherKX0 is nef,

• or there exists a rational curveCi as in the theorem. This curve cannot be exceptional, henceX0 is eitherP² or a ruled surface, and the original surface X has a morphism to a smooth curve whose generic fiber is P¹. Starting from a given surface X of this type, there are several possible different end productsX0(see Exercise 5.7.1)b)).

In particular,if X is not birational to a ruled surface, it has a minimal model X0 with KX0 nef. We prove that this model is unique.

Proposition 5.22 LetX andY be smooth projective surfaces and letπ:X 99K Y be a birational map. IfKY is nef, πis a morphism. If both KX andKY are nef,π is an isomorphism.

Proof. Let ε : ˜Y → Y be the blow-up of a point and let Ce ⊂ Ye be an integral curve other than the exceptional curveE, with imageC⊂Y. We have ε^∗C∼^linCe+mE for some m≥0 andK_Ye =ε^∗KY +E. Therefore,

(K_Ye·C) = (Ke Y ·C) +m≥(KY ·C)≥0.

There is by Theorem 5.18 a (minimal) composition of blow-upsε: ˜X →Xsuch that ˜π=π◦εis a morphism, itself a composition of blow-ups by Theorem 5.20.

Ifεis not an isomorphism, its last exceptional curve E is not contracted by ˜π

hence must satisfy, by what we just saw, (K_Xe·E)≥0. But this is absurd since

this integer is−1. henceπis a morphism.

5.7 Exercises

1) Letπ:X →B be a ruled surface.

a) Let ˜X→X be the blow-up a pointx. Describe the fiber of the composi-tion ˜X →X →B overπ(x).

b) Show that the strict transform in ˜X of the fiberπ⁻¹(π(x)) can be con-tracted to give another ruled surfaceX(x)→B.

c) LetFn be a Hirzebruch surface (withn∈N; see Example 5.7). Describe the surfaceFn(x) (Hint: distinguish two cases according to whetherxis on the curveCn of Example 5.7).

2) LetX be a projective surface and letD andH be Cartier divisors onX. a) AssumeH is ample, (D·H) = 0, andD6∼^num0. Prove (D²)<0.

b) Assume (H²)>0. Prove the inequality (Hodge Index Theorem) (D·H)²≥(D²)(H²).

When is there equality?

c) Assume (H²)>0. IfD1, . . . , Drare divisors onX, settingD0=H, prove (−1)^rdet((Di·Dj))0≤i,j≤r≥0.

3) LetD1, . . . , Dnbe nef Cartier divisors on a projective varietyXof dimension n. Prove

(D1·. . .·Dn)ⁿ ≥(Dⁿ₁)·. . .·(D_nⁿ).

(Hint: first do the case when the divisors are ample by induction on n, using Exercise 2)b) whenn= 2).

4) LetKbe the function field of a curve over an algebraically closed field, and let X be a subscheme of P^N_K defined by homogeneous equations f1, . . . , fr of respective degreesd1, . . . , dr. Ifd1+· · ·+dr≤N, show thatX has aK-point (Hint: proceed as in the proof of Theorem 5.2).

5) (Weil) Let C be a smooth projective curve over a finite field Fq, and let F : C → C be the Frobenius morphism obtained by taking qth powers (it is

indeed an endomorphism ofC becauseC is defined overFq). LetX =C×C, let ∆⊂X be the diagonal (see Example 5.1), and let Γ⊂X be the graph ofF.

a) Compute (Γ²) (Hint: proceed as in Example 5.1).

b) Letx1 andx2 be the respective classes of{?} ×C andC× {?}. For any divisorD onX, prove

(D²)≤2(D·x1)(D·x2) (Hint: apply Exercise 2)c) above).

c) SetN = Γ·∆. Prove

|N−q−1| ≤2g√q

(Hint: apply b) torΓ +s∆, for allr, s∈Z). What does the numberN count?

6) Show that the group of automorphisms of a smooth curveC of genus g≥2 is finite (Hint: consider the graph Γ of an automorphism of C in the surface X=C×C, show that (KX·Γ) is bounded, and use Example 5.1 and Theorem 4.8.b)).

Chapter 6

Parametrizing morphisms

We concentrate in this chapter on basically one object, whose construction dates back to Grothendieck in 1962: the space parametrizing curves on a given variety, or more precisely morphisms from a fixed projective curveC to a fixed smooth quasi-projective variety. Mori’s techniques, which will be discussed in the next chapter, make systematic use of these spaces in a rather exotic way.

We will not reproduce Grothendieck’s construction, since it is very nicely explained in [G2] and only the end product will be important for us. However, we will explain in some detail in what sense these spaces areparameter spaces, and work out their local structure. Roughly speaking, as in many deformation problems, the tangent space to such a parameter space at a point isH⁰(C,F), whereF is some locally free sheaf onC, first-order deformations are obstructed by elements ofH¹(C,F), and the dimension of the parameter space is therefore bounded from below by the differenceh⁰(C,F)−h¹(C,F). The crucial point is that sinceChas dimension 1, this difference is the Euler characteristic ofF, which can be computed from numerical data by the Riemann-Roch theorem.

6.1 Parametrizing rational curves

Letkbe a field. Anyk-morphismf from P¹_k toP^N_k can be written as

f(u, v) = (F0(u, v), . . . , FN(u, v)), (6.1) where F0, . . . , FN are homogeneous polynomials in two variables, of the same degreed, with no nonconstant common factor ink[U, V] (or, equivalently, with no nonconstant common factor in ¯k[U, V], where ¯kis an algebraic closure ofk).

We are going to show that there exist universal integral polynomials in the coefficients of F0, . . . , FN which vanish if and only if they have a nonconstant

common factor in ¯k[U, V], i.e., a nontrivial common zero in P¹k¯. By the

is surjective, hence of rankm+ 1 (herek[U, V]mis the vector space of homoge-neous polynomials of degreem). This map being linear and defined overk, we conclude that F0, . . . , FN have a nonconstant common factor ink[U, V] if and only if, for allm, all (m+ 1)-minors of some universal matrix whose entries are linear integral combinations of the coefficients of theFi vanish. This defines a Zariski closed subset of the projective spaceP((Sym^dk²)^N+1),defined overZ.

Therefore, morphisms of degree d from P¹ to P^N are parametrized by a Zariski open set of the projective spaceP((Sym^dk²)^N+1); we denote this quasi-projective variety Mord(P¹,P^N). Note that these morphisms fit together into auniversal morphism

f^univ: P¹×Mord(P¹,P^N) −→ P^N (u, v), f

7−→ F0(u, v), . . . , FN(u, v) . Finally, morphisms fromP¹ toP^N are parametrized by the disjoint union

Mor(P¹,P^N) = G

d≥0

Mord(P¹,P^N) of quasi-projective schemes.

Let nowXbe a (closed) subscheme ofP^N defined by homogeneous equations G1, . . . , Gm. Morphisms of degree d from P¹ to X are parametrized by the subscheme Mord(P¹, X) of Mord(P¹,P^N) defined by the equations

Gj(F0, . . . , FN) = 0 for all j ∈ {1, . . . , m}.

Again, morphisms fromP¹ toX are parametrized by the disjoint union Mor(P¹, X) = G

d≥0

Mord(P¹, X)

of quasi-projective schemes. The same conclusion holds for any quasi-projective variety X: embed X into some projective varietyX; there is a universal mor-phism

f^univ:P¹×Mor(P¹, X)−→X

and Mor(P¹, X) is the complement in Mor(P¹, X) of the image by the (proper) second projection of the closed subscheme (f^univ)⁻¹(X X).

If nowX can be defined by homogeneous equationsG1, . . . , Gmwith coeffi-cients in a subringRof k, the scheme Mord(P¹, X) has the same property. If

m is a maximal ideal of R, one may consider the reduction Xm of X modulo m: this is the subscheme of P^N_R/m defined by the reductions of the Gj mod-ulo m. Because the equations defining the complement of Mord(P¹,P^N) in P((Sym^dk²)^N+1) are defined overZand the same for all fields, Mord(P¹, Xm) is the reduction of theR-scheme Mord(P¹, X) modulo m. In fancy terms, one may express this as follows: if X is a scheme over SpecR, the R-morphisms P¹_R→X are parametrized by (theR-points of) a locally noetherian scheme

Mor(P¹_R,X)→SpecR

and the fiber of a closed pointmis the space Mor(P¹,Xm).