Sharp volume and multiplicity bounds for Fano simplices

TL;DR

This paper establishes precise upper bounds on volume, Mahler volume, and multiplicity for Fano simplices based on dimension and Gorenstein index, and provides an explicit classification method for these simplices.

Contribution

It introduces sharp bounds for Fano simplices and an efficient classification procedure applicable to any dimension and Gorenstein index.

Findings

Derived sharp bounds on volume, Mahler volume, and multiplicity.

Classified Fano simplices up to dimension four for various Gorenstein indices.

Developed an explicit classification algorithm for Fano simplices.

Abstract

We present sharp upper bounds on the volume, Mahler volume and multiplicity for Fano simplices depending on the dimension and Gorenstein index. These bounds rely on the interplay between lattice simplices and unit fraction partitions. Moreover, we present an efficient procedure for explicitly classifying Fano simplicies of any dimension and Gorenstein index and we carry out the classification up to dimension four for various Gorenstein indices.