Secant varieties of toric varieties arising from simplicial complexes

TL;DR

This paper develops a combinatorial framework to analyze secant varieties of toric embeddings from simplicial complexes, focusing on Segre-Veronese varieties and their singularities.

Contribution

It proves that all such secant varieties are toric and classifies those that are Gorenstein or $ ext{Q}$-Gorenstein, with explicit singular locus descriptions.

Findings

All secant varieties considered are toric.

Complete classification of Gorenstein and $ ext{Q}$-Gorenstein secants.

Explicit description of the singular locus.

Abstract

Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes. We prove that every such secant is toric, which gives a way to use combinatorial tools to study singularities. We focus on the Segre-Veronese variety for which we completely classify their secants that give Gorenstein or $\mathbb Q$-Gorenstein varieties. We conclude providing the explicit description of the singular locus.