Week 7. Traditional minimal models and Castelnuovo's theorem
characterising rational surfaces by pg = P2 = q = 0.
Beauville's CUP book Complex algebraic surfaces gives a more
traditional treatment of classification of surfaces in comparatively
simple style. Also, lots of examples and exercises.
Blowups, birational maps
From the theory of DVRs, we know that a rational map
f: C - -> X from a nonsingular curve C to a projective variety X is a
morphism -- is regular at every point P in C. (Or extends to a morphism
in a unique way.) If follows that two nonsingular projective curves
C1, C2 are birational iff they are isomorphic.
For surfaces, blowups and blowdowns contradict this: if P in S is a
nonsingular point of a surface, the blowup si: S1 -> S of P with
exceptional curve E a -1-curve, is a birational morphism (an isomorphism
outside the exceptional locus E -> P). An overall conclusion is that the
ambiguity in choosing a minimal model of surfaces consists of blowups
and blowdowns. This is an approximate slogan, and there is a lot more to
say.
Facts. It is known (and proved in what follows) that
(1) Castelnuovo's contractibility criterion: a -1-curve E in S is
contracted by the inverse of a blowdown si: S -> S1.
(2) Every surface can be blown down to a model with no -1-curves.
(3) Resolution of indeterminacy: given a rational map f: S - -> X to
a projective variety, there is a chain of blowups
Sn .. -> Si -> .. S0 -> S
so that the composite rational map f: Sn -> S - -> X is a morphism.
(4) A birational morphism S -> T (regular at every point of S) between
nonsingular projective surfaces is a composite of blowdowns of -1-curves.
(3+4) A birational map S - -> T fits in a house diagram
S <- Y -> T where the two morphism forming the roof are composites
of blowdowns.
The contractibility criterion (1) is proved (in the more general context
of Artin rational singularities) in my Park City Chapters, Prop 4.15.
For a more modest proof, see Beauville's book. Resolution of
indeterminacies is proved by an intersection numbers argument (making a
blowup of a point of indeterminacy decreases degree by at least 1).
The proof is in Beauville's book.
Before Mori theory, a _minimal model_ always meant a surface with no
-1-curves. A final conclusion is that a surface either has K nef, or is
PP^2, or is a PP^1-bundle over a nonsingular curve. This was of course
understood as a distant consequence in ancient times, but with the
emphasis on birational ideas, it seems that it never came to the fore
until Mumford's first paper on the classification. Nowadays we mostly
reserve the word minimal (in strict treatment of Mori theory) for
surfaces with K nef (and PP^2 or FF_n or the other ruled surfaces
with no -1-curves are described as Mori fibre spaces (Mfs).
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Ruled surfaces, Tsen's theorem, minimal ruled surfaces over B as
PP(E) for E a rank 2 vector on B.
In ancient times, classification was viewed in terms of rationality:
given V, can it be parametrised by rational functions? For example,
Lueroth's theorem: if a curve C over k = kbar is the image of a dominant
rational map PP^1 - -> C then g(C) = 0 and C is birational to PP^1. A
similar result holds for surfaces, but the result is harder, and only
works in characteristic zero. The ancients understood that this fails in
higher dimensions, and in the 1960s, with new technology, three different
3-fold counterexamples were given, with arguments and rigorous proofs.
If there is a rational curve C through every general point P of a
variety V in char 0, then V has no canonical forms: Roughly, the
assumption lead to a dominant rational map B x PP^1 - -> V with B
lower dimensional, and then s in Ga(V,KV) must vanish on the images
of the lines PP^1, and so s=0. This highlights the case of ruled
surfaces.
A _ruled surface_ is a surface S with a rational map f: S - -> B to
a base curve with general fibre a rational curve (a _ruling_). The
generic fibre of f is a rational curve over the function field k(B)
of the base.
Lemma If g(B) >= 1 then f is a morphism.
Proof Otherwise, make a series of blowups to resolve the
indeterminacies of f. The exceptional curve of the final blowup
must dominate B. Then by Lueroth's theorem, g(B) = 0.
Tsen's theorem
Over an algebraically closed field k = kbar, a ruled surface
f: S - -> B has a _section_, a curve E in S that maps
isomorphically to the base curve B. It follows that S is
birational to PP^1 x Bin a way that is compatible with f.
There are 2 quite different proofs. Beauville, Chap III gives the
topological proof over CC. PoincarĂ© duality in topology says a
section exists exists in cohomology, and it can be obtained as an
algebraic cycle by the exponential sequence.
On the other hand, the arithmetic of PP^1 over a C1 field gives a
proof from a completely different point of view. The generic fibre C
is a rational curve defined over k(B). Therefore it has a model as a
plane conic C in PP^2, provided by its anticanonical class -KC.
Claim. Let B be a curve defined over an algebraically closed
field k and set L = k(B). Let C be a curve over L with g(C)=0
(that is, C is rational over Lbar). Then C is rational over L.
Write B for an affine model, and k[B] for its affine coordinate
ring, with a choice of coordinates. Then k[B]{<=n} grows linearly
in n.
Proof The main point: C is isomorphic over the nonclosed field L to
a plane conic. C2 in PP^2. In fact, go to Lbar. The definition of
rational is that C iso PP^1 over Lbar, so it has a rational point P
over Lbar. Then the canonical class KC = O(-2P) so the divisor class
2P is invariant under Gal(Lbar/L). Therefore the linear system
Lbar(2P) defines an embedding C into PP^2 that is invariant under
Gal(Lbar/L), so is defined over L itself.
The equation of C is a quadratic form q(x,y,z) with q in L.
Treat L iso k(B) = Frac(k[B]). If we take x,y,z to be polynomials
in t of degree <= n, we get 3*n + (a bit) free coefficients in k.
On the other hand, q(x,y,z) is a quadratic form. We can clear
denominators to get its coefficients in k[t], say polynomials of
degree <= d. Then for chosen polynomials x,y,z of degree <= n,
q(x,y,z) has degree d + 2n, so imposing the condition q(x,y,z) = 0
is at most d + 2n + 1 coefficients. This proves the claim.
(Neither of these proofs work over a higher dimensional base. We get
into the Brauer group H^2(OX^x) as obstruction.)
Beauville's book does ruled surfaces in great detail, and relates to
VBAC. Special case of g(B)=0 when every vb is O(a)+O(b) and g(B)=1 when
there are stable and sstable that are nonsplit.
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Week 7, Lecture 3
Castelnuovo's characterisation of rational surfaces.
Mostly taken from Beauville's book, Chap V.
Theorem Suppose S is a nonsingular surface with no -1-curves
and q = P2 = 0. (Here q is the irregularity q = h^1(OS),
and P2 the n=2 plurigenus Pn = h^0(S,nKS).)
(I) There exists an effective divisor D with KD < 0 and
h^0(OS(K+D)) = 0 (also written as |K+D| = emptyset).
(II) S contains an irreducible curve C with pa(C) = 0 and C^2 > 0
(pa(C) = 0 means C is a nonsingular rational curve, or C = PP^1).
(III) S is rational.
Both implications (I) => (II) => (III) are straightforward.
For take D as in (I). Some irreducible component C of D has
KC < 0 and |K+C| = emptyset. RR gives
chi(K+C) = 1 + 1/2(C+K)C = pa(C).
So C is a nonsingular rational curve with KC < 0, and since
-1-curves are excluded, C^2 >= 0, which is (II).
RR gives that C moves in a nontrivial linear system |C|. A
pencil {C,D} from that gives a ruling on S, with C iso PP^1
as a fibre, and Tsen's theorem implies (III).
===
The proof of (I) breaks up into several cases. The
easiest is
Case K^2 = 0. Then chi(-K) = chi(2k) = 1, so that -K lineq effective
divisor. Now for H a hyperplane section -KH = -K(H+nK) > 0. But
since -K is effective, there is a value n with
D in |H+nK| but |H+(n+1)K| = emptyset.
This proves (I) in this case.
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Next case K^2 < 0. As a first step in the proof of (I), we find an
effective divisor E with KE < 0. Let H be ample. First, if KH <= 0
then E in |K+nH| for every n > 0 has KE < 0.
If KH > 0 we use instead a threshold argument. Set r0 = KH/(-K)^2
so that
(H + r0K)K = 0 and (H + r0K)^2 > 0.
Choose p,q >> 0 with p/q = r0 + ep (small rational) and use RR
for qH+pK. Since RR has a quadratic term 1/2D^2, we get
chi(qH+pK) = 1/2*q^2*(H+rK)^2 + lower order terms
and this goes to infinity as const.*q^2 (independently of ep). (This
is the same argument as in my Chap. D -- the quadratic term in RR
dominates; see also Beauville, Lemma V.8.)
Given effective E with KE < 0, some component C of E also has
KC < 0. Now C^2 < 0 is excluded by no -1-curves, so C^2 >= 0.
But now |aC + nK| = emptyset for any a and n >> 0 (because it
has intersection number (aC+nK)C < 0 whereas C^2 >= 0).
This gives (I)
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Final case K^2 > 0. RR and P2 = 0 implies that chi(-K) >= 2.
If |-K| has a reducible element A+B then KA < 0 say, and
A+K = -B is not effective, so taking D = A gives (I).
From now on, we use K = -D is an irreducible divisor with
D^2 > 0, so that DC >= 0 for every curve on S.
For every effective divisor H, there exist n > 0 with
E in |H+nK| and |E+K| = emptyset.
Defer discussion of the special case
(*) E = 0 for every choice of H, that is, every effective divisor
is linearly equivalent to a multiple of -K.
Then E = sum niCi is effective and nonzero. Now -KC = DC >= 0 for
every C = Ci. Now |K+C| = emptyset, so pa(C) = 0 and every C is a
nonsingular rational curve. As before, C^2 = -1 is excluded.
If KC <= -2, then C^2 >= 0 giving (I). If KC = 0 then some element
of |-K| intersects C, so must contain it, contradicting the
assumption that |-K| has no reducible elements.
==
Finally (*) is not possible (in char 0 using topology). Every
effective divisor a multiple of -K implies that K or -K generates
Pic S. Then by the exponential sequence its class also generates
H^2(S,ZZ), so K^2 = 1. This contradicts Noether's formula
chi(OS) = 1 = 1/12(K^2 + c2).
This can also be done in char p, but not today.
Notice that if B2 = 1 then S is PP^2, so -KS = O(3).
==
Compare Castelnuovo's proof (+ Kodaira) in Beauville, Chap V
with the Mori treatment in my Chap D and E.
The classical proof sets out from global analytic properties
(the invariants q = h^1(OS) and Pn = h^0(nKS)). It uses from
the outset the constant term in RR
chi(OS) = 1 - q + pg (which = 1 in this argument)
= 1/12(c1^2 +c2)
It then runs into curves with KC < 0 and "adjunction
terminates" as items in the argument. It also runs into
the nef threshold r0 as a rational number, as in my Chap D.
Mori theory starts with the question of K not nef, and deduces the
existence of an extremal ray, which gives either -1-curve, or a
PP^1-bundle structure, or S = PP^2. It separates the discussion into
the obstruction to K nef in numerical terms of N^1 and NEbar, and
only starts discussing plurigeneral and other global analytic
properties once the minimal model is established.