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. 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 $\mathscr L^{m,n}_{r,k}$, [12] provides its irreducible decomposition. We further provide a specific polynomial family defining its irreducible components.