Degrees of high-dimensional subvarieties of determinantal varieties

Let n = p(alpha)b, where p is a prime, and g.c.d. (p, b) = 1. In p(n2-1) let X-r be the variety defined by rank((x(i,j))) less than or equal to n - r. We show that any subvariety of X-r of codimension less than p(alpha)r 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.