2014/10/07 by Wheeler, Ashley K.
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1410.1910
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 \mathfrak Pt denote the ideal generated by the size t principal minors of X. When t=2 the resulting quotient ring K[X]/\mathfrak 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 \mathfrak Pt and \mathfrak Pn-t in the localized polynomial ring, where we invert det X, are isomorphic. From that we show the algebraic set given by \mathfrak 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.