Difference between revisions of "Kawamata rationality theorem"
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− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Y. Kawamata, K. Matsuda, K. Matsuki, "Introduction to the minimal model problem" , ''Algebraic Geometry (Sendai 1985)'' , ''Adv. Stud. Pure Math.'' , '''10''' , Kinokuniya& North-Holland (1987) pp. 283–360 {{MR|0946243}} {{ZBL|0672.14006}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Y. Kawamata, "On the length of an extremal rational curve" ''Invent. Math.'' , '''105''' (1991) pp. 609–611 {{MR|1117153}} {{ZBL|0751.14007}} </TD></TR></table> | + | <table> |
+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> Y. Kawamata, K. Matsuda, K. Matsuki, "Introduction to the minimal model problem" , ''Algebraic Geometry (Sendai 1985)'' , ''Adv. Stud. Pure Math.'' , '''10''' , Kinokuniya& North-Holland (1987) pp. 283–360 {{MR|0946243}} {{ZBL|0672.14006}} </TD></TR> | ||
+ | <TR><TD valign="top">[a2]</TD> <TD valign="top"> Y. Kawamata, "On the length of an extremal rational curve" ''Invent. Math.'' , '''105''' (1991) pp. 609–611 {{MR|1117153}} {{ZBL|0751.14007}} </TD></TR> | ||
+ | </table> | ||
+ | |||
+ | [[Category:Algebraic geometry]] |
Revision as of 17:19, 18 October 2014
A theorem stating that there is a strong restriction for the canonical divisor of an algebraic variety to be negative while the positivity is arbitrary. It is closely related to the structure of the cone of curves and the existence of rational curves.
Definitions and terminology.
Let be a normal algebraic variety (cf. Algebraic variety). A
-divisor
on
is a formal linear combination of a finite number of prime divisors
of
with rational number coefficients
(cf. also Divisor). The canonical divisor
is a Weil divisor on
corresponding to a non-zero rational differential
-form for
(cf. also Differential form). The pair
is said to be weakly log-terminal if the following conditions are satisfied:
The coefficients of satisfy
.
There exists a positive integer such that
is a Cartier divisor (cf. Divisor).
There exists a projective birational morphism from a smooth variety such that the union
![]() |
is a normal crossing divisor (cf. Divisor), where is the strict transform of
and
coincides with the smallest closed subset
of
such that
is an isomorphism.
One can write
![]() |
such that for all
.
There exist positive integers such that the divisor
is
-ample (cf. also Ample vector bundle).
For example, the pair is weak log-terminal if
is smooth and
is a normal crossing divisor, or if
has only quotient singularities and
.
Rationality theorem.
Let be a normal algebraic variety defined over an algebraically closed field of characteristic
, and let
be a
-divisor on
such that the pair
is weakly log-terminal. Let
be a projective morphism (cf. Projective scheme) to another algebraic variety
, and let
be an
-ample Cartier divisor on
. Then (the rationality theorem, [a1])
![]() |
is either or a rational number. In the latter case, let
be the smallest positive integer such that
is a Cartier divisor, and let
be the maximum of the dimensions of geometric fibres of
. Express
for relatively prime positive integers
and
. Then
.
For example, equality is attained when ,
,
is a point, and
is a hyperplane section.
The following theorem asserts the existence of a rational curve, a birational image of the projective line , and provides a more geometric picture. However, the estimate of the denominator
obtained is weaker: In the situation of the above rationality theorem, if
, then there exists a morphism
such that
is a point and
[a2].
The two theorems are related in the following way: If , then
is no longer
-ample. However, there exists a positive integer
such that the natural homomorphism
![]() |
![]() |
is surjective for any positive integer (the base-point-free theorem, [a1]). Let
be the associated morphism over the base space
. Then any positive dimensional fibre of
is covered by a family of rational curves as given in the second theorem [a2].
References
[a1] | Y. Kawamata, K. Matsuda, K. Matsuki, "Introduction to the minimal model problem" , Algebraic Geometry (Sendai 1985) , Adv. Stud. Pure Math. , 10 , Kinokuniya& North-Holland (1987) pp. 283–360 MR0946243 Zbl 0672.14006 |
[a2] | Y. Kawamata, "On the length of an extremal rational curve" Invent. Math. , 105 (1991) pp. 609–611 MR1117153 Zbl 0751.14007 |
Kawamata rationality theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kawamata_rationality_theorem&oldid=23874