A numerical ampleness criterion via Gale duality

TL;DR

This paper develops a numerical criterion for determining when Weil divisors on $ ext{Q}$-factorial complete toric varieties admit positive multiples that are either nef or ample, using Gale duality and secondary fans.

Contribution

It introduces a Gale duality-based numerical criterion for ampleness and nefness of divisors on toric varieties, including computation of Cartier indices and characterizations of special Fano varieties.

Findings

Provides a method to compute Cartier index of Weil divisors.

Offers a numerical characterization of various Fano toric varieties.

Includes multiple examples illustrating the criteria and computations.

Abstract

The main object of the present paper is a numerical criterion determining when a Weil divisor of a $\Q$--factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is a consequence of $\Z$--linear interpretation of Gale duality and se\-con\-dary fan as developed in several previous papers of us. As a byproduct we get a computation of the Cartier index of a Weil divisor and a numerical characterization of weak $\Q$--Fano, $\Q$--Fano, Gorenstein, weak Fano and Fano toric varieties. Several examples are then given and studied. \keywords{$\Q$--factorial complete toric variety \and ample divisor \and nef divisor \and $\Z$-liner Gale duality \and secondary fan \and ampleness criterion \and Cartier index \and $\Q$-Fano toric variety.