Geometry of some determinantal arrangements

Abstract: Let $D_{m,n}\subset\C^{m\times n}$ be the union of the zero loci of the maximal minors of a generic $m\times n$ matrix, and let $U_{m,n}$ be its complement. We show that $U_{m,n}\cong\GL_m(\C)\times\Gr(m,n)<sup>\circ$, where $\Gr(m,n)<sup>\circ$ is the open stratum of the Grassmannian on which all Plücker coordinates are nonzero. For $m\in{2,n-2}$ this gives closed formulas for the topology of $U_{m,n}$ and recovers results of Yim. Our main result, which is computer-assisted, is that $D_{3,5}$ is \emph{not} a free divisor. This answers a question of Yim. We determine $\Der(-\log D_{3,5})$ explicitly. It has projective dimension one, and it fails to be free exactly on the matrices of rank at most two. There the failure comes from a non-free transversal arrangement: the ten planes spanned by pairs of five general vectors in $\C<sup>3$. Computations in further cases support the conjecture that $D_{m,n}$ is free only for $m\le 2$ or $n=m+1$.