Fine Polyhedral Adjunction Theory

TL;DR

This paper introduces a modified version of polyhedral adjunction theory based on the Fine interior of lattice polytopes, leading to a better-behaved framework with stronger results and simpler proofs.

Contribution

It proposes the Fine adjoint polytope, enhancing polyhedral adjunction theory with improved properties and extending existing results with more natural proofs.

Findings

Enhanced properties of Fine adjoint polytopes

Decomposition of polytopes into Cayley sums

Finiteness of the Fine spectrum

Abstract

Originally introduced by Fine and Reid in the study of plurigenera of toric hypersurfaces, the Fine interior of a lattice polytope got recently into the focus of research. It is has been used for constructing canonical models in the sense of Mori Theory [arXiv:2008.05814]. Based on the Fine interior, we propose here a modification of the original adjoint polytopes as defined in [arXiv:1105.2415], by defining the Fine adjoint polytope $P^{F(s)}$ of $P$ as consisting of the points in $P$ that have lattice distance at least $s$ to all valid inequalities for $P$. We obtain a Fine Polyhedral Adjunction Theory that is, in many respects, better behaved than its original analogue. Many existing results in Polyhedral Adjunction Theory carry over, some with stronger conclusions, as decomposing polytopes into Cayley sums, and most with simpler, more natural proofs as in the case of the finiteness of the Fine spectrum.