On singularities of determinantal hypersurfaces

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.