Socle degrees for local cohomology modules of thickenings of maximal minors and sub-maximal Pfaffians

Let $S$ be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let $I$ be the determinantal ideal of maximal minors or $\operatorname{Pf}$ the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu--Weyman--Witt to determine the $S$-module structures of $\operatorname{Ext}^j_S(S/I^t, S)$ and $\operatorname{Ext}^j_S(S/\operatorname{Pf}^t, S)$, from which we get the degrees of generators of these $\operatorname{Ext}$ modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang on the socle degrees of local cohomology modules of the form $H^j_\mathfrak{m}(S/I^t)$.Comment: Final version. Comments welcome

Mixed Hodge structure on local cohomology with support in determinantal varieties

We employ the inductive structure of determinantal varieties to calculate the weight filtration on local cohomology modules with determinantal support. We show that the weight of a simple composition factor is uniquely determined by its support and cohomological degree. As a consequence, we obtain the equivariant structure of the Hodge filtration on each local cohomology module, and we provide a formula for its generation level. In the case of square matrices, we express the Hodge filtration in terms of the Hodge ideals for the determinant hypersurface. As an application, we describe a recipe for calculating the mixed Hodge module structure on any iteration of local cohomology functors with determinantal support.Comment: 17 pages, comments welcom