Determinantal Varieties Over Truncated Polynomial Rings

We study higher order determinantal varieties obtained by considering generic $m\times n$ ($m \le n$) matrices over rings of the form $F[t]/(t^k)$, and for some fixed $r$, setting the coefficients of powers of $t$ of all $r \times r$ minors to zero. These varieties can be interpreted as generalized tangent bundles over the classical determinantal varieties; a special case of these varieties first appeared in a problem in commuting matrices. We show that when $r = m$, the varieties are irreducible, but when $r < m$, these varieties have at least $\lfloor {k/2}\rfloor + 1$ components. In fact, when $r=2$ (for any $k$), or when $k=2$ (for any $r$), there are exactly $\lfloor {k/2}\rfloor + 1$ components. We give formulas for the dimensions of these components in terms of $k$, $m$, and $n$. In the case of square matrices with $r=m$, we show that the ideals of our varieties are prime and that the coordinate rings are complete intersection rings, and we compute the degree of our varieties via the combinatorics of a suitable simplicial complex.