Minimal model theorem for toric divisors

Abstract

Minimal model conjecture for a proper variety $X$ is that if $\kappa (X)\ge 0$, 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 $\kappa (X)\ge 0$ 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.

AMS-Latex, text 14 pages, figures 2 pages. The figures are not submitted because of a technical reason. A person who wants the figure pages is asked to contact to the Author. She will send a hard copy of the figures by postal mail

References in corpus (1)

• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties