Hodge ideals for the determinant hypersurface

Abstract

We determine explicitly the Hodge ideals for the determinant hypersurface as an intersection of symbolic powers of determinantal ideals. We prove our results by studying the Hodge and weight filtrations on the mixed Hodge module $\mathcal {O}_{\mathscr {X}}(*\mathscr {Z})$ of regular functions on the space $\mathscr {X}$ of $n\times n$ matrices, with poles along the divisor $\mathscr {Z}$ of singular matrices. The composition factors for the weight filtration on $\mathcal {O}_{\mathscr {X}}(*\mathscr {Z})$ are pure Hodge modules with underlying $\mathcal {D}$-modules given by the simple ${\text {GL}}$-equivariant $\mathcal {D}$-modules on $\mathscr {X}$, where ${\text {GL}}$ is the natural group of symmetries, acting by row and column operations on the matrix entries. By taking advantage of the ${\text {GL}}$-equivariance and the Cohen–Macaulay property of their associated graded, we describe explicitly the possible Hodge filtrations on a simple ${\text {GL}}$-equivariant $\mathcal {D}$-module, which are unique up to a shift determined by the corresponding weights. For non-square matrices, $\mathcal {O}_{\mathscr {X}}(*\mathscr {Z})$ is replaced by the local cohomology modules $H^{\bullet }_{\mathscr {Z}}(\mathscr {X},\mathcal {O}_{\mathscr {X}})$, which turn out to be pure Hodge modules. By working out explicitly the Decomposition Theorem for some natural resolutions of singularities of determinantal varieties, and using the results on square matrices, we determine the weights and the Hodge filtration for these local cohomology modules.