Optimal upper bounds for anti-canonical volumes of singular toric Fano varieties

TL;DR

This paper establishes optimal upper bounds for the anti-canonical volumes of certain singular toric Fano varieties and related polytopes, providing explicit bounds and constructions to demonstrate their optimality.

Contribution

It introduces the first sharp bounds for anti-canonical volumes of $rac{1}{q}$-lc toric Fano varieties and constructs examples to show these bounds are tight.

Findings

Derived explicit upper bounds for anti-canonical volumes.

Constructed examples achieving the bounds, proving their optimality.

Provided bounds for volumes of lattice simplices with a single interior lattice point.

Abstract

Fix two positive integers $d\geq3$ and $q$. We give an upper bound for anti-canonical volumes of $d$-dimensional $\frac{1}{q}$-lc toric Fano varieties, which corresponds to an upper bound for the dual normalized volumes of the associated $d$-dimensional $\frac{1}{q}$-lc Fano polytopes. And we also construct examples to show that these upper bounds are optimal. Besides, we provide an optimal upper bound for volumes of $d$-dimensional lattice simplices $S$ such that $\frac{1}{q}S$ has exactly one interior lattice point.