Toric double determinantal varieties

TL;DR

This paper studies toric double determinantal varieties, proving their irreducibility and smoothness, deriving a dimension formula, and using a specific example to explore an open problem in local algebra.

Contribution

It provides an elementary proof of irreducibility and smoothness for toric double determinantal varieties, along with a dimension formula and empirical insights into an open problem.

Findings

Toric double determinantal varieties are irreducible and smooth.

A straightforward formula for their dimension is established.

Empirical evidence is provided for an open problem in local algebra.

Abstract

We examine Li's double determinantal varieties in the special case that they are toric. We recover from the general double determinantal varieties case, via a more elementary argument, that they are irreducible and show that toric double determinantal varieties are smooth. We use this framework to give a straighforward formula for their dimension. Finally, we use the smallest nontrivial toric double determinantal variety to provide some empirical evidence concerning an open problem in local algebra.