Diagonal splittings of toric varieties and unimodularity

TL;DR

The paper studies when toric varieties X admit diagonally compatible splittings, using a polyhedral criterion governed by the vector configuration Sigma(1) of the fan. It proves a complete dichotomy: Sigma(1) being unimodular implies X is diagonally split for all q >= 2, while failure of 2-regularity rules out diagonal splitting for any q, with nuanced behavior when Sigma(1) is 2-regular but not unimodular, especially in low dimensions. Hermite normal form is used to analyze 2-regularity and its relation to torsion in N mod the span of Sigma(1), yielding partial results and sharp examples (notably in dimension 4) that illustrate the limits of these criteria. The paper also corrects and extends to compatibly split subdiagonals the theory of diagonal splittings in products X^n, with implications for projective normality and Koszulness, and includes detailed treatments of root-system-like configurations and the Birkhoff polytope as instructive cases.

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.

Theorems & Definitions (26)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• proof : Proof of Theorem \ref{['thm:unimodular']}
• Proposition 3.1
• proof