Hilbert series of second order jets of determinantal varieties

In this paper, we will investigate the jet schemes of determinantal varieties. It is quite often the case that the geometric information concerning the jet schemes of an algebraic variety can be described, but the more refined algebraic information is quite mysterious. For example, it is known that computing the Hilbert function associated to a natural grading on these jet schemes is a very hard problem. The present paper handles a few such computations. It succeeds in computing the Hilbert functions of the second order jet schemes in the case of maximal minors of a $2\times n$ matrix.