On The Existence Of Flips
Hacon and McKernan’s paper, arxiv alg-geom/0507597
Brian Lehmann, February 2007
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Introduction
References: Hacon and McKernan’s paper, as above. Kollár and Mori,
Birational Geometry of Algebraic Varieties
What is the minimal model program? Looking for some good choice of
birational model of smooth projective varieties. The term “minimal model”
has a technical meaning, but we’re going to ignore that for the moment.
Brief verbal review of surfaces: structure of birational maps, Castelnuovo’s theorem
One way to generalize is to note that −1-curves have negative intersection with the canonical class KX . A higher-dimension generalization is the
cone theorem, which says that we can contract curves for which C.KX is
negative. Our goal: reduce to either KX nef, or a mori fiber space. A Mori
Fiber Space is a map X → Z to a space of smaller dimension, of relative
Picard number one, such that −KX is relatively ample. (Relative Picard
number is: dimension of vector space of all irreducible curves contracted by
map, up to numerical equivalence).
Compare to situation with surfaces
These varieties have special properties, i.e. positivity of KX or having
a fibration with Fano fibers, and it’s our hope that they will give us a
better understanding of their birational equivalence class. In general we
can’t expect a unique ending point, but we hope to understand the different
possibilities as well.
Why do we expect that we can do this? If KX is not nef, then by
definition there is an irreducible curve C with C.KX < 0. The cone theorem
allows us to find a map contracting C.
In particular, we come up with a map f : X → Z of relative picard
number one, with −KX relatively ample. There are several possibilities:
1. The dimension of Z is less than that of X
2. dim(X) = dim(Z), f is birational and the exceptional locus contains
a divisor
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3. dim(X) = dim(Z), f is birational and the exceptional locus has codimension at least 2
We want to show that, no matter which category we fall in, we are closer
to our goal than before.
If we have case 1, then X → Z is a Mori fiber space, so we are done.
Otherwise, the map is birational. Say that it contracts a divisor. It is not
hard to show that the exceptional locus is in fact a divisor, so we contract
nothing else. Now Z may be singular, but it turns out that we can get a
handle on these types of singularities. In particular, we know that KZ is still
Q-Cartier, so we can replace X by Z and continue to run the program. Note
that we have removed an obstruction to the nefness of KX . So, we hope that
we are “closer” to a nef divisor than before. In fact, by contracting a divisor,
we have reduced the picard number, so we know this type of contraction will
have to stop at some point, which will leave us in one of the other cases.
Finally, the map may contract something of codimension at least 2. In
parallel to the previous case, we hope that we can somehow end up with
a variety with canonical class “more nef” than before. It is this case that
gives us the most trouble.
Proposition 1.1 Suppose X is a normal proj. Q-factorial variety, and
f : X → Z is the contraction as in case 3. Then KZ is not Q-Cartier.
Proof: If it were, say mKZ is a Cartier divisor, then we would have
f ∗ (mKZ ) = mKX , except possibly over the contracted subset. Since this
subset has codimension at least 2, these two divisors are actually equal.
However, let C be a curve contracted by f . We have C.KX < 0 and
C.f ∗ (KZ ) = 0, a contradiction. 2
So, in case 3, we can no longer run the MMP on the resulting variety Z,
since it no longer makes sense to ask whether KZ is nef or not. Flips are
defined to give us a handle on this case.
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Preliminaries
Now, we start defining things more precisely.
Definition 2.1 A log pair (X, ∆) is a pair consisting of a normal variety
X and a Q-Weil divisor ∆, such that KX + ∆ is Q-Cartier. We call the
pair Q-factorial if X is.
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Suppose we have a birational map g : Y → X, with Y normal. Say that
g −1 : X → Y is defined on the open set U . For any prime Weil divisor D
−1
on X, we define g∗−1 (D) to be the closure
P of the image of D|U under g .
In particular, we can extend to ∆ = P
(ai Di ) in the natural way. Then,
we have KY + g∗−1 (∆) = g ∗ (KX + ∆) + E a(E, ∆)E for some exceptional
divisors E. This is because the two sides are isomorphic away from the
exceptional locus.
These values of a(E, ∆) give some measure of how singular ∆ is. We will
deal with two types of singularities of divisors. A divisor (X, ∆) is purely
log terminal if the infimum of such a(E, ∆) is greater than −1. A divisor
(X, ∆) is Kawamata log terminal if it is purely log terminal, and in addition
b∆c is 0. Although it seems like these conditions are very hard to confirm,
they only really need to be checked on a good log resolution. In particular,
∆ is KLT if and only if its multiplier ideal is all of OX .
If you’re not familiar with these notions, you should think of them as
giving some bound on how singular the divisor can be. For example, if ∆
is purely log terminal, then b∆c is a disjoint union of irreducible normal
components. These aren’t every type of singularity that comes up in the
MMP, but these are sufficient for the purposes of this talk.
Definition 2.2 A small contraction is a birational map f : X → Z, such
that f has connected fibers and the exceptional locus has codimension at least
two.
The following definition has some minor variants between different authors.
Definition 2.3 A flipping contraction for a Q-Cartier divisor D on a normal X is a projective birational map f : X → Z such that
1. Z is normal
2. f is a small contraction
3. −D is relatively ample
If in addition f has relative picard number 1, the contraction is called extremal.
Note that our case 3 above meets all of these criteria. So, we’re really
going to be interested in constructing flips for KX or, more generally, KX +
∆, where ∆ is a slightly singular divisor.
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Definition 2.4 Suppose f : X → Z is a flipping contraction for D. Then
the flip of D, if it exists, is the diagram with X 0 normal, X → X 0 birational,
and the strict transform D0 of D is relatively ample and Q-Cartier. We also
require that X 0 → Z be small.
Conceptually speaking, we are doing the following: instead of contracting
the exceptional locus, we remove it, and replace it with something which
does not have negative intersection with the canonical divisor. So, rather
than passing to Z and running the MMP, we have constructed a space X 0
which we hope has KX 0 more nef than before, and we continue running the
program there. One can prove that if such an X 0 exists, it shares many of
the same properties as X.
Proposition 2.5 Let (X, ∆) be a projective, Q-factorial klt pair, and g :
X → Z a (flipping) contraction of a (KX + ∆)-negative extremal ray. Suppose that the flip X 0 → Z exists. Then 1) X 0 is Q-factorial, and the picard
numbers of X and X 0 are the same. 2) The pair (X 0 , ∆0 ) is also klt.
Proof: For 1), we’ll need to use a corollary of the cone theorem: Suppose
R is the extremal ray which we are contracting. Let C ⊂ X be a curve
generating R. Then we have an exact sequence
0 → Pic(Z) → Pic(X) → Z
Where the first map is L → g ∗ L and the second is M → (M.C).
Now, since φ : X → X 0 is an isomorphism in codimension 1, it induces an
isomorphism of Weil divisors. If D0 is a Weil divisor on X 0 , corresponding to
D on X. Then for some rational number r we have R.(D + r(KX + ∆)) = 0.
By the above, there is a Cartier divisor DZ on Z such that
m(D + r(KX + ∆)) ∼ g ∗ DZ
So, mD0 = mφ∗ (D) ∼ (g 0 )∗ DZ − mr(KX 0 + ∆0 ) is Q-Cartier.
I won’t prove 2 - the general idea is to take a common resolution of X
and X 0 and show the positivity of certain coefficients using the Hodge Index
Theorem. 2
The most important questions regarding flips are existence and termination.
Existence Conjecture: Let (X, ∆) be a kawamata log terminal pair, with
X Q-factorial. Suppose f : X → Z is a flipping contraction for KX + ∆.
Then the flip of KX + ∆ exists.
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Termination conjecture: Let (X, ∆) be a kawamata log terminal pair,
with X Q-factorial. Then there is no infinite sequence of KX + ∆ flips.
We can make essentially the same conjectures with R-divisors, if we
change our definitions slightly. We restrict ourselves to Q-factorial X, since
one can show that this case implies similar results for non-Q-factorial X.
If these two conjectures are true, then it’s clear that the MMP will in
fact give us our goal. The goal of this paper is to show that existence and
termination in dimension (n − 1) leads to existence of flips in dimension n.
A warning: although we work almost exclusively with Q-divisors, we need
the strength of the conjectures for R-divisors to run the induction. In fact,
one can show that existence of flips for Q-divisors implies that of R-divisors.
This is not the best result known; the recent paper by Birkar Cascini
Hacon McKernan shows the existence of a canonical model for pairs (X, ∆)
with KX + ∆ big. In particular this implies the existence of some kind of
minimal models for smooth projective varieties. So, why bother with this
paper? Well, it introduces many of the ideas that used, and it’s still an
important result.
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Basic Properties of Flips
What is the general strategy of showing the existence of flips? There are
two fundamental propositions that we will need.
Proposition 3.1 Let Z be a normal variety, and D a Weil divisor on Z.
Then the following are equivalent:
1. ⊕m≥0 OZ (mD) is finitely generated.
2. There exists a projective small contraction g : Y → Z such that Y is
normal, and the Weil divisor D0 = g∗−1 (D) is Q-Cartier and relatively
ample.
In this case, there is a unique g : Y → Z satisfying condition 2.
Corollary 3.2 Suppose that (X, ∆) is a klt, Q-factorial pair, and that f :
X → Z is a flipping contraction for KX + ∆. Let r ∈ Z>0 be such
that r(KX + ∆) is a Cartier divisor. Then, the flip exists if and only if
⊕m≥0 f∗ OX (mr(KX + ∆)) is a finitely-generated sheaf of OZ ) algebras. If
the flip exists, it is unique.
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Proof: Assume 1. By replacing D by some multiple, we may assume that
OZ (D) generates ⊕m≥0 OZ (mD). Thus, we may define Y = ProjZ ⊕m≥0
OZ (mD).
We need to check the properties given in 2. The natural g : Y → Z is
birational; in fact, it is an isomorphism on any open set where D is principal.
It’s clear that Y is normal, and g∗−1 (D) is the twisting sheaf OY (1).
We just need to show that g : Y → Z is a small contraction. Suppose that
there were an exceptional divisor E. We would have OY ( OY (E). Then for
some sufficiently large m, OZ (mD) = g∗ OY (m) ( g∗ (OY (m)(E)). However,
OZ (mD) is reflexive, i.e. it is the dual of its dual. If a reflexive sheaf
and some other sheaf are isomorphic on any open set whose complement
has codimension at least 2, then they are isomorphic everywhere. These
two sheaves are isomorphic away from the image g(Ex(g)). This set has
codimension at least 2, giving us our contradiction.
Conversely, we first show that OZ (mD) = g∗ OY (mD0 ). Note that
g∗ OY (mD0 ) → OZ (mD) is an injection: Since g∗−1 (D) ⊂ g ∗ (D), we have
g∗ (g∗−1 (D)) ⊂ g∗ g ∗ (D) = D. For surjectivity, note a section of OZ (mD) on
an open set U ⊂ Z lifts to a section of OY (mD0 ) on g −1 (U ) − Ex(g). Since
the exceptional locus has codimension at least 2, we can extend the section
to all of g −1 (U ). Thus the map is also surjective.
Now, since D0 is relatively ample, ⊕m≥0 OZ (mD) is finitely generated
(this is just the relative version of the usual fact that the ring of sections of
an ample line bundle is finitely generated). Furthermore, Y = Proj ⊕m≥0
OZ (mD), showing that Y is unique. (This is the relative version of the
familiar fact from Hartshorne). 2
Because of the local nature of the construction, we will usually assume
that Z is affine. Another important idea is the following reduction, due to
Shokurov:
Definition 3.3 Let f : X → Z be a flipping contraction for KX + ∆. We
say that this is a pl flipping contraction (for prelimiting) if in addition
1. X is Q-factorial
2. (X, ∆) is plt
3. S = b∆c is irreducible, and −S is relatively ample
Shokurov also introduced an idea called “special termination”, a special
case of termination of flips. I don’t want to get into the exact definition now
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- the only thing we need to know for this paper is that “special termination”
in dimension n is implied by the existence and termination conjectures in
dimension n − 1.
Theorem 3.4 To prove the existence of flips for KLT pairs in dimension
n, it suffices to construct flips for pl flipping contractions and prove special
termination in dimension n.
This idea is useful, because it opens the door for an inductive argument.
Specifically, the existence of an irreducible divisor S = b∆c allows us to
relate dimension n to smaller dimensions.
At last, we’ve come to a point where we can understand the gist of Hacon
and McKernan’s proof. They aim to show that pl flips exist in dimension
n, assuming existence and termination of flips in lower dimensions. They
will prove this by showing that the appropriate algebra is finitely generated.
They prove this by induction on dimension, reducing the question to finite
generation of a restricted algebra on the divisor S. More explicitly, they:
1. Construct a log resolution Y → X under which the transform T of S
has nice properties.
2. Find a good birational model T 0 of T , using the induction hypothesis
of the MMP in dimension n − 1.
3. Show that some algebra on T is finitely generated, using the relationship between T 0 and T .
4. Show that this is equivalent to finite generation of the algebra we
started with.
Although the proofs of these steps are subtle, they are not too long.
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