Rational points of fixed denominator in real toric arrangements

TL;DR

This paper establishes conditions under which rational points of fixed denominator exist in real toric arrangements and explores implications for the structure sheaf's Frobenius pushforward on smooth toric varieties.

Contribution

It provides a new sufficient condition linking rational points in toric arrangements to the decomposition of Frobenius pushforwards in toric geometry.

Findings

Identifies a sufficient condition on m for rational points in each stratum

Connects these conditions to the summands in the Frobenius pushforward

Enhances understanding of rational points in real toric arrangements

Abstract

We give a sufficient condition on a positive integer $m$ for every stratum of a given real toric hyperplane arrangement to contain a rational point of denominator $m$. As a consequence, we give a sufficient condition on $m$ for the degree $m$ Frobenius pushforward of the structure sheaf on a smooth toric variety to contain all possible summands in the Picard group.