Canonical models of toric hypersurfaces

TL;DR

This paper constructs canonical models of nondegenerate toric hypersurfaces using the Fine interior of their Newton polytopes, revealing their Kodaira dimension and fiber structure through combinatorial methods.

Contribution

It introduces a canonical model construction for toric hypersurfaces based on the Fine interior, linking geometric properties to combinatorial data.

Findings

Kodaira dimension equals the minimum of d-1 and the dimension of F(P)

The general fibers are nondegenerate toric hypersurfaces of Kodaira dimension 0

Provides a combinatorial formula for the intersection number of the canonical divisor

Abstract

Let $Z$ be a nondegenerate hypersurface in $d$-dimensional torus $(\mathbb{C}^*)^d$ 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 $F(P) \neq \emptyset$ we construct a unique projective model $\widetilde{Z}$ of $Z$ having at worst canonical singularities and obtain minimal models $\hat{Z}$ of $Z$ by crepant morphisms $\hat{Z}\to \widetilde{Z}$. We show that the Kodaira dimension $\kappa =\kappa(\widetilde{Z})$ equals $\min \{ d-1, \dim F(P) \}$ and the general fibers in the Iitaka fibration of the canonical model $\widetilde{Z}$ 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 $(K_{\widetilde{Z}})^{d-1}$.