The plurigenera of nondegenerate minimal toric hypersurfaces

TL;DR

This paper provides a general formula for the plurigenera of minimal models of nondegenerate toric hypersurfaces in any dimension, linking these invariants to lattice points on the Fine interior, and explores their canonical models.

Contribution

It introduces a new formula for plurigenera of toric hypersurfaces based on lattice points, applicable in arbitrary dimensions, and discusses their canonical models.

Findings

Derived a formula for plurigenera in terms of lattice points

Established a relation between the volume of the canonical divisor and lattice points

Provided methods to construct canonical models under certain conditions

Abstract

In this article we present a formula for the plurigenera of minimal models of nondegenerate toric hypersurfaces, which is valid in arbitrary dimension and which expresses these invariants through lattice points on the Fine interior. From this formula we derive a formula for the volume of the canonical divisor of the toric hypersurface. We also study the pluricanonical mappings and show under some restrictions how to construct a canonical model of the toric hypersurface.