The plurigenera of nondegenerate toric hypersurfaces

Abstract

Let $Z_f\subset ({\mathbb{C}}^{\ast }{)}^n$ be a toric hypersurface defined by a nondegenerate Laurent polynomial $f$ with the Newton polytope $\Delta$ and let $X$ be a projective birational model of $Z_f$ with at worst canonical singularities. We present a combinatorial formula for all plurigenera $P_m(X)$ which is valid in arbitrary dimension $n\ge 2$. This solves a problem posed by Reid in 1987. Our formula for $P_m(X)$ is based on counting lattice points in $m$-th and $(m-1)$-th dilates of some rational subpolytope $F(\Delta )\subset \Delta$ introduced by Fine. We prove this formula using minimal models $Y$ of $Z_f$ and apply it to compute $K_Y^{n-1}$ and to study the pluricanonical mappings when $Y$ is of general type.