Ideals Generated by Principal Minors

Original · EN

A minor is principal means it is defined by the same row and column indices. Let X be a square generic matrix, K[X] the polynomial ring in entries of X, over an algebraically closed field, K. For fixed t≤ n, let Pₜ denote the ideal generated by the size t principal minors of X. When t=2 the resulting quotient ring K[X]/P₂ is a normal complete intersection domain. When t>2 we break the problem into cases depending on a fixed rank, r, of X. We show when r=n for any t, the respective images of Pₜ and Pₙ₋ₜ in the localized polynomial ring, where we invert X, are isomorphic. From that we show the algebraic set given by Pₙ₋₁ has a codimension n component, plus a codimension 4 component defined by the determinantal ideal (which is given by all the submaximal minors of X). When n=4 the two components are linked, and we prove some consequences.