Frobenius splittings of toric varieties

TL;DR

This paper explores Frobenius splittings in toric varieties, providing a polyhedral criterion for diagonal splitting, and examines implications for section rings and Schubert varieties.

Contribution

It introduces a characteristic-free Frobenius splitting criterion for toric varieties and analyzes its consequences for algebraic properties and Schubert varieties.

Findings

Section rings of nef line bundles are normally presented and Koszul on diagonally split toric varieties.

Schubert varieties are generally not diagonally split.

A polyhedral criterion for diagonal splitting in toric varieties.

Abstract

We discuss a characteristic free version of Frobenius splittings for toric varieties and give a polyhedral criterion for a toric variety to be diagonally split. We apply this criterion to show that section rings of nef line bundles on diagonally split toric varieties are normally presented and Koszul, and that Schubert varieties are not diagonally split in general.