Kawamata rationality theorem

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