Canonical models of toric hypersurfaces

Abstract

Let $Z \subset \mathbb{T}_d$ be a non-degenerate hypersurface in $d$-dimensional torus $\mathbb{T}_d \cong (\mathbb{C}^*)^d$ defined by a Laurent polynomial $f$ with a given $d$-dimensional Newton polytope $P$. It follows from a theorem of Ishii that $Z$ is birational to a smooth projective variety $X$ of Kodaira dimension $\kappa \geq 0$ if and only if the Fine interior $F(P)$ of $P$ is nonempty. We define a unique projective model $\widetilde{Z}$ of $Z$ having at worst canonical singularities which allows us to obtain minimal models $\widehat{Z}$ of $Z$ by crepant morphisms $\widehat{Z} \to \widetilde{Z}$. Moreover, we show that $\kappa = \min \{ d-1, \dim F(P) \}$ and that 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 the rational polytope $F(P)$, we compute the stringy $E$-function of minimal models $\widehat{Z}$ and obtain a combinatorial formula for their stringy Euler numbers.