On jet schemes of determinantal varieties

TL;DR

This paper explores the jet schemes of determinantal varieties, revealing new connections with Hilbert series and establishing shellability and irreducible decompositions for specific cases.

Contribution

It introduces novel links between determinantal varieties' jet schemes and their Hilbert series, including shellability proofs and explicit irreducible component descriptions.

Findings

Established a connection between Hilbert series and jet schemes of determinantal varieties.

Proved shellability of certain jet schemes for special cases.

Provided explicit irreducible decompositions and defining polynomials for jet schemes.

Abstract

Determinantal varieties are important objects of study in algebraic geometry. In this paper, we will investigate them using the jet scheme approach. We have found a new connection for the Hilbert series between a determinantal variety and its jet schemes. We denote the $k$-th order jet scheme of the determinantal variety defined by $r$-minors in an $m \times n$ matrix as $\mathscr{L}^{m,n}_{r,k}$. For the special case where $m$, $n$, and $r$ are equal, and $m$ and $r$ are 3 while $k$ is 1, we establish a correspondence between the defining ideals of $\mathscr{L}^{m,n}_{r,k}$ and abstract simplicial complexes, proving their shellability and obtaining the Hilbert series of $\mathscr{L}^{m,n}_{r,k}$ accordingly. Moreover, for general $\mathscr{L}^{m,n}_{r,k}$, \cite{12} provides its irreducible decomposition. We further provide a specific polynomial family defining its irreducible components. Keywords. Determinantal varieties, jet schemes, shellability, Hilbert series.