Initial Complex Associated to a Jet Scheme of a Determinantal Variety

TL;DR

This paper proves that the principal component of the first order jet scheme over a determinantal variety of matrices of rank at most 1 is arithmetically Cohen-Macaulay, using combinatorial shellability techniques.

Contribution

It establishes the Cohen-Macaulay property of a specific jet scheme component for determinantal varieties, a novel result in algebraic geometry.

Findings

Principal component of the jet scheme is arithmetically Cohen-Macaulay.

Associated Stanley-Reisner complex is shellable.

Provides combinatorial proof of Cohen-Macaulayness.

Abstract

We show in this paper that the principal component of the first order jet scheme over the classical determinantal variety of m x n matrices of rank at most 1 is arithmetically Cohen-Macaulay, by showing that an associated Stanley-Reisner simplicial complex is shellable.