Symbolic powers of the generic linkage of maximal minors

Abstract: Let $I$ be the ideal generated by the maximal minors of a matrix $X$ of indeterminates over a field and let $J$ denote the generic link, i.e., the most general link, of $I$. The generators of the ideal $J$ are not known. We provide an explicit description of the lead terms of the generators of $J$ using Gr\"obner degeneration: For a carefully chosen term order, the reduced Gr\"obner basis of the generic link $J$ is a minimal set of its generators and the initial ideal of $J$ is squarefree. We leverage this description of the initial ideal to establish the equality of the symbolic and ordinary powers of $J$. Our analysis of the initial ideal readily yields the Gorenstein property of the associated graded ring of $J$, and, in positive characteristic, the $F$-rationality of the Rees algebra of $J$. Using the technique of $F$-split filtrations, we further obtain the $F$-regularity of the blowup algebras of $J$.