Minimal model theorem for toric divisors

Details Minimal model theorem for toric divisors Minimal model theorem for toric divisors

1997-05-29

arxiv.org/abs/alg-geom/9705026v1

Mathematics

Algebraic Geometry

AMS-Latex, text 14 pages, figures 2 pages. The figures are not submitted because of a technical reason. A person who wants the

Scientific paper

Minimal model conjecture for a proper variety $X$ is that if $\kappa(X)\geq 0$, then $X$ has a minimal model with the abundance and if $\kappa =-\infty$, then $X$ is birationally equivalent to a variety $Y$ which has a fibration $Y \to Z$ with $-K_Y$ relatively ample. In this paper, we prove this conjecture for a $\D$-regular divisor on a proper toric variety by means of successive contractions of extremal rays and flips of ambient toric variety. Furthermore, for such a divisor $X$ with $\kappa(X)\geq 0$ we construct a projective minimal model with the abundance in a different way; by means of "puffing up" of the polytope, which gives an algorithm of a construction of a minimal model.

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