Lattice 3-polytopes of lattice width 2 and corresponding toric hypersurfaces

TL;DR

This paper classifies certain 3-dimensional lattice polytopes with a specific interior structure, leading to new examples of toric hypersurfaces of general type with bounded genus.

Contribution

It provides a classification of 3D lattice polytopes with 2D Fine interiors and up to 40 interior points, enriching the understanding of associated toric hypersurfaces.

Findings

Classified all 2D Fine interiors of 3D lattice polytopes with ≤40 interior points.

Constructed many examples of surfaces of general type with genus ≤40.

Connected lattice polytope properties to the Kodaira dimension of hypersurfaces.

Abstract

The Kodaira dimension of a nondegenerate toric hypersurface can be computed from the dimension of the Fine interior of its Newton polytope according to recent work of Victor Batyrev, where the Fine interior of the Newton polytope is the subpolytope consisting of all points which have an integral distance of at least $1$ to all integral supporting hyperplanes. In particular, if we have a Fine interior of codimension $1$, then the hypersurface is of general type and the Newton polytope has lattice width $2$. In this article we study this situation for lattice $3$-polytopes and the corresponding surfaces of general type. In particular, we classify all $2$-dimensional Fine interiors of those lattice $3$-polytopes which have at most $40$ interior lattice points, thus obtaining many examples of surfaces of general type and genus at most $40$.