Degrees of high-dimensional subvarieties of determinantal varieties

Abstract:

Let $n = p^ab$, where $p$ is a prime, and $\text {g.c.d. }(p,b)=1$. In $\mathbf {P}^{n^2-1}$, let $X_r$ be the variety defined by $\text {rank} ((x_{i,j})) \le n-r$. We show that any subvariety of $X_r$ of codimension less than $p^ar$ must have degree a multiple of $p$. We also show that the bounds on the codimension in our results are strict by exhibiting subvarieties of the appropriate codimension whose degrees are prime to $p$.