Diagonal splittings of toric varieties and unimodularity

TL;DR

This paper characterizes when toric varieties are diagonally split using polyhedral criteria, linking the property to unimodularity and 2-regularity of the primitive generator configuration.

Contribution

It provides a polyhedral criterion for diagonal splittings of toric varieties, connecting the property to unimodularity and 2-regularity of the defining vector configuration.

Findings

X is diagonally split at all q iff the configuration is unimodular.

X is not diagonally split at any q if the configuration is not 2-regular.

Implications for the set of q where X is diagonally split.

Abstract

We use a polyhedral criterion for the existence of diagonal splittings to investigate which toric varieties X are diagonally split. Our results are stated in terms of the vector configuration given by primitive generators of the 1-dimensional cones in the fan defining X. We show, in particular, that X is diagonally split at all q if and only if this configuration is unimodular, and X is not diagonally split at any q if this configuration is not 2-regular. We also study implications for the possibilities for the set of q at which a toric variety X is diagonally split.