On the Fine Interior of Three-dimensional Canonical Fano Polytopes

TL;DR

This paper investigates the Fine interior of three-dimensional canonical Fano polytopes, providing computational insights crucial for constructing minimal birational models of certain hypersurfaces.

Contribution

It offers the first comprehensive computational analysis of the Fine interior for all 3D canonical Fano polytopes, totaling 674,688 cases.

Findings

Computed the Fine interior for all 3D canonical Fano polytopes.

Identified properties of the Fine interior relevant to birational geometry.

Provided data useful for constructing minimal models of hypersurfaces.

Abstract

The Fine interior $\Delta^{\text{FI}}$ of a $d$-dimensional lattice polytope $\Delta$ is a rational subpolytope of $\Delta$ which is important for constructing minimal birational models of non-degenerate hypersurfaces defined by Laurent polynomials with Newton polytope $\Delta$. This paper presents some computational results on the Fine interior of all $674,\!688$ three-dimensional canonical Fano polytopes.