Finiteness of the polyhedral Q-codegree spectrum

TL;DR

This paper proves that for a certain class of lattice polytopes, the set of Q-codegrees above any positive number is finite, confirming a conjecture in the context of Q-Gorenstein toric varieties.

Contribution

It establishes the finiteness of the Q-codegree spectrum for lattice polytopes with -canonical normal fans, including Q-Gorenstein cases, thus confirming Fujita's Spectrum Conjecture in this setting.

Findings

Finiteness of the Q-codegree spectrum above positive thresholds.

Includes Q-Gorenstein normal fans of index r.

Proves Fujita's Spectrum Conjecture for Q-Gorenstein toric varieties.

Abstract

In this paper we show that the spectrum of the Q-codegree of a d-dimensional lattice polytope is finite above any positive threshold in the class of lattice polytopes with \alpha-canonical normal fan for any fixed \alpha>0. For \alpha=1/r this includes lattice polytopes with Q-Gorenstein normal fan of index r. In particular, this proves Fujita's Spectrum Conjecture for polarized varieties in the case of Q-Gorenstein toric varieties of index r.