Principal minor ideals and rank restrictions on their vanishing sets

All matrices we consider have entries in a fixed algebraically closed field K. A minor of a square matrix is principal means it is defined by the same row and column indices. We study the ideal generated by size t   principal minors of a generic matrix, and restrict our attention to locally closed subsets of its vanishing set, given by matrices of a fixed rank. The main result is a computation of the dimension of the locally closed set of n×nn×n rank n−2n−2 matrices whose size n−2n−2 principal minors vanish; this set has dimension n2−n−4n2−n−4.