Diagonal splittings of toric varieties and unimodularity

Abstract

We use a polyhedral criterion for the existence of diagonal splittings to investigate which toric varieties X 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 X . We show, in particular, that X X is diagonally split at all q q if and only if this configuration is unimodular, and that X X is not diagonally split at any q q if this configuration is not 2 2 -regular. We also study implications for the possibilities for the set of q q at which a toric variety X X is diagonally split.