# Gr\"obner bases and the Cohen-Macaulay property of Li's double determinantal varieties

## TL;DR

This paper proves Li's conjecture that double determinantal varieties are Cohen-Macaulay, normal, and irreducible, by using liaison theory and provides a formula for their dimension.

## Contribution

It establishes the Cohen-Macaulay property and normality of double determinantal varieties, extending previous results and offering a dimension formula.

## Key findings

- Double determinantal varieties are Cohen-Macaulay, normal, and irreducible.
- The defining ideals have a Gröbner basis given by natural generators.
- A formula for the dimension of these varieties is provided.

## Abstract

We consider double determinantal varieties, a special case of Nakajima quiver varieties. Li conjectured that double determinantal varieties are normal, irreducible, Cohen-Macaulay varieties whose defining ideals have a Gr\"obner basis given by their natural generators. We use liaison theory to prove this conjecture in a manner that generalizes results for mixed ladder determinantal varieties. We also give a formula for the dimension of a double determinantal variety.