Mori theory of extremal rays
Let be a projective morphism of algebraic varieties over a field
of characteristic
(cf. also Algebraic variety). A relative
-
-cycle is a formal linear combination
of a finite number of curves
(reduced irreducible
-dimensional closed subschemes) on
with real number coefficients
such that
are points on
. (If
, then the word "relative" is dropped.) Two relative
-
-cycles
and
are said to be numerically equivalent if their intersection numbers are equal,
for any Cartier divisor
on
(cf. also Divisor; Intersection index (in algebraic geometry)). The set
of all the equivalence classes of relative
-
-cycles with respect to the numerical equivalence becomes a finite-dimensional real vector space. The closed cone of curves (the Kleiman–Mori cone)
is defined to be the closed convex cone in
generated by the classes of curves on
which are mapped to points on
by
. A half-line
is called an extremal ray if the inequality
holds and if the equality
for
implies
.
Contents
Cone theorem.
Let be a normal algebraic variety and
an effective
-divisor such that the pair
is weakly log terminal (cf. Kawamata rationality theorem). Let
be a projective morphism to another algebraic variety. Then there exist at most countably many extremal rays
(
) satisfying the following conditions:
For any , there exist an element
and numbers
, which are zero except for finitely many
, such that
and
.
(discreteness) For any closed convex cone in
such that
for any
, there exist only finitely many
such that
.
Contraction theorem.
Let be an extremal ray as above. Then there exists a morphism
, called a contraction morphism, to a normal algebraic variety
with a morphism
which is characterized by the following properties:
;
;
any curve which is mapped to a point by
is mapped to a point by
if and only if its numerical class belongs to
.
Two methods of proofs for the cone theorem are known. The first one [a6] uses a deformation theory of morphisms over a field of positive characteristic and applies only in the case where is smooth. It is important to note that this is the only known method in mathematics to prove the existence of rational curves (as of 2000). The second approach [a2] uses a vanishing theorem of cohomology groups (cf. Kawamata–Viehweg vanishing theorem) which is true only in characteristic
. This method of proof, which is obtained via a rationality theorem (cf. Kawamata rationality theorem), applies also to singular varieties and easily extends to the logarithmic version as explained above. The contraction theorem has been proved only by a characteristic-
method (cf. [a1]).
In the following it is also assumed that the variety is
-factorial, that is, for any prime divisor
on
there exists a positive integer
, depending on
, such that
is a Cartier divisor. Then the contraction morphism
is of one of the following types:
(Fano–Mori fibre space) .
(divisorial contraction) There exists a prime divisor of
such that
and
induces an isomorphism
.
(small contraction) is an isomorphism in codimension
, in the sense that there exists a closed subset
of codimension
of
such that
induces an isomorphism
.
Flip conjectures.
The first flip conjecture is as follows: Let be a small contraction. Then there exists a birational morphism from a
-factorial normal algebraic variety
which is again an isomorphism in codimension
and is such that the pair
with
is weakly log terminal and
is a
-ample
-divisor (cf. also Divisor). The diagram
is called a flip (or log flip). Note that
is
-ample.
The second flip conjecture states that there does not exist an infinite sequence of consecutive flips.
There is no small contraction if . The flip conjectures have been proved for
(see [a3], [a4] for the first flip conjecture, and [a5], [a7] for the second). The proofs depend on the classification of singularities and it is hard to extend them to a higher-dimensional case.
Minimal model program (MMP).
Fix a base variety and consider a category whose objects are a pair
and a projective morphism
such that
is a
-factorial normal algebraic variety and
is a
-divisor such that
is weakly log terminal. A morphism from
to
in this category is a birational mapping
which is surjective in codimension
, in the sense that any prime divisor on
is the image of a prime divisor on
, and such that
and
. The minimal model program is a program which works under the assumption that the flip conjectures hold. It starts from an arbitrary object
and constructs a morphism to another object
such that one of the following holds:
has a Fano–Mori fibre space structure
over
.
is minimal over
in the sense that
is
-nef, i.e., an inequality
holds for any curve
on
such that
is a point on
. Construct objects
inductively as follows. Set
. Suppose that
has already been constructed. If
is
-nef, then a minimal model is obtained. If not, then, by the cone theorem, there exists an extremal ray and one obtains a contraction morphism
by the contraction theorem. If
, then a Fano–Mori fibre space is obtained. If
is a divisorial contraction, then one sets
. If
is a small contraction and if the first flip conjecture is true, then take the flip
and set
. If the second flip conjecture is true, then this process stops after a finite number of steps.
A normal algebraic variety is said to be terminal, or it is said that
has only terminal singularities, if the following conditions are satisfied:
1) The canonical divisor is a
-Cartier divisor.
2) There exists a projective birational morphism from a smooth variety with a normal crossing divisor
such that one can write
with
for all
.
As a special case of the minimal model program, if one assumes that has only terminal singularities and
, then any subsequent pair satisfies the same condition that
has only terminal singularities and
. This is the "non-log" version.
It is expected that the minimal model program works also over a field of arbitrary characteristic, although the cone and contraction theorems are conjectural in general.
References
[a1] | Y. Kawamata, K. Matsuda, K. Matsuki, "Introduction to the minimal model problem" Adv. Stud. Pure Math. , 10 (1987) pp. 283–360 |
[a2] | Y. Kawamata, "The cone of curves of algebraic varieties" Ann. of Math. , 119 (1984) pp. 603–633 |
[a3] | S. Mori, "Flip theorem and the existence of minimal models for 3-folds" J. Amer. Math. Soc. , 1 (1988) pp. 117–253 |
[a4] | V. Shokurov, "3-fold log flips" Izv. Russian Akad. Nauk. , 56 (1992) pp. 105–203 |
[a5] | V. Shokurov, "The nonvanishing theorem" Izv. Akad. Nauk. SSSR , 49 (1985) pp. 635–651 |
[a6] | S. Mori, "Threefolds whose canonical bundles are not numerically effective" Ann. of Math. , 116 (1982) pp. 133–176 |
[a7] | Y. Kawamata, "Termination of log-flips for algebraic 3-folds" Internat. J. Math. , 3 (1992) pp. 653–659 |
Mori theory of extremal rays. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mori_theory_of_extremal_rays&oldid=17891