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 ${\mathcal{L}}_{r,k}^{m,n}$. 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 ${\mathcal{L}}_{r,k}^{m,n}$ and abstract simplicial complexes, proving their shellability and obtaining the Hilbert series of ${\mathcal{L}}_{r,k}^{m,n}$ accordingly. Moreover, for general ${\mathcal{L}}_{r,k}^{m,n}$, \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.