On singularities of determinantal hypersurfaces

TL;DR

The paper investigates singularities of determinantal hypersurfaces defined by maximal minors of a matrix of regular functions and relates them to the incidence varieties formed with projective space. It uses arc-space methods to translate questions about log canonical thresholds and rational singularities of $(X,Z_A)$ into codimension estimates of contact loci for $(Y,W_A)$, establishing sharp inequalities and an equivalence in the square case. It then applies these results to configuration hypersurfaces, proving that configuration incidence varieties have rational singularities when the configuration matroid is connected, and provides smoothness criteria via 1-generic matrices. Overall, the work builds a bridge between determinantal geometry, arc-theoretic invariants, and matroid theory with implications for positive characteristic via $F$-rational-type phenomena.

Abstract

Given a closed subscheme $Z$ in a smooth variety $X$, defined by the maximal minors of an $s\times r$ matrix of regular functions, with $s\geq r$, we consider the corresponding incidence correspondence $W$ in $Y=X\times {\mathbf P}^{r-1}$, and relate the log canonical thresholds of $(X,Z)$ and $(Y,W)$. In particular, when $r=s$, we show that ${\rm lct}(X,Z)=1$ if and only if ${\rm lct}(Y,W)=r$. Moreover, in this case, we show that $Z$ has rational singularities if and only if $W$ has pure codimension $r$ in $Y$ and has rational singularities. As a consequence, we deduce that for a configuration hypersurface with a connected configuration matroid, the corresponding configuration incidence variety has rational singularities.

Theorems & Definitions (19)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Proposition 2.4
• proof
• proof : Proof of Theorem \ref{['thm1_main']}
• proof : Proof of Theorem \ref{['thm2_main']}