Canonical models of toric hypersurfaces

Abstract

Let $Z$ be a nondegenerate hypersurface in $d$-dimensional torus $({\mathbb{C}}^{\ast }{)}^d$ defined by a Laurent polynomial $f$ with a $d$-dimensional Newton polytope $P$. The subset $F(P)\subset P$ consisting of all points in $P$ having integral distance at least $1$ to all integral supporting hyperplanes of $P$ is called the Fine interior of $P$. If $F(P)\ne \varnothing$ we construct a unique projective model $\tilde{Z}$ of $Z$ having at worst canonical singularities and obtain minimal models $\hat{Z}$ of $Z$ by crepant morphisms $\hat{Z}\to \tilde{Z}$. We show that the Kodaira dimension $\kappa =\kappa (\tilde{Z})$ equals $\min \{d-1,\dim F(P)\}$ and the general fibers in the Iitaka fibration of the canonical model $\tilde{Z}$ are non\-degenerate $(d-1-\kappa )$-dimensional toric hypersurfaces of Kodaira dimension $0$. Using $F(P)$, we obtain a simple combinatorial formula for the intersection number $(K_{\tilde{Z}}{)}^{d-1}$.

41 pages, final version, to appear in Algebraic Geometry Compositio Mathematica