On jet schemes of determinantal varieties

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. By $\mathscr{L}^{m,n}_{r,k}$, we denote the $k$th order jet scheme of the determinantal variety defined by $r$-minors in an $m \times n$ matrix. For the special case where $m$, $n$, and $r$ are equal, and $m$ and $r$ are both 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 $\mathscr{L}^{m,n}_{r,k}$, Docampo [11] provides its irreducible decomposition. We further provide a specific polynomial family defining its irreducible components.