Ideals generated by principal minors

Abstract

A minor is principal means it is defined by the same row and column indices. We study ideals generated by principal minors of size $t\leq n$ of a generic $n\times n$ matrix $X$, in the polynomial ring generated over an algebraically closed field by the entries of $X$. When $t=2$ the resulting quotient ring is a normal complete intersection domain. We show for any $t$, upon inverting $\det X$ the ideals given respectively by the size $t$ and the size $n-t$ principal minors become isomorphic. From that we show the algebraic set given by the size $n-1$ principal minors has a codimension $4$ component defined by the determinantal ideal, plus a codimension $n$ component. When $n=4$ the two components are linked, and in fact, geometrically linked.