Hilbert series of second order jets of determinantal varieties

TL;DR

This paper investigates the algebraic structure of second order jet schemes of determinantal varieties, successfully computing their Hilbert functions in specific cases involving maximal minors of a 2 by n matrix.

Contribution

It provides explicit Hilbert function computations for second order jet schemes of determinantal varieties, a problem previously considered very challenging.

Findings

Computed Hilbert functions for second order jet schemes of maximal minors of 2×n matrices

Enhanced understanding of algebraic properties of jet schemes in determinantal varieties

Demonstrated methods to handle complex algebraic computations in jet schemes

Abstract

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.