On singularities of determinantal hypersurfaces

TL;DR

This paper investigates the singularities of determinantal hypersurfaces, establishing relationships between their log canonical thresholds and rational singularities through incidence correspondences.

Contribution

It provides new criteria linking the singularities of determinantal hypersurfaces and their incidence varieties, especially when the matrix sizes are equal.

Findings

${ m lct}(X,Z)=1$ iff ${ m lct}(Y,W)=r$ when $r=s$

$Z$ has rational singularities iff $W$ has rational singularities and pure codimension $r$

Configuration incidence varieties with connected matroids have rational singularities

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.