On the codimension of permanental varieties

In this article, we study permanental varieties, i.e. varieties defined by the vanishing of permanents of fixed size of a generic matrix. Permanents and their varieties play an important, and sometimes poorly understood, role in combinatorics. However, there are essentially no geometric results about them in the literature, in very sharp contrast to the well-behaved and ubiquitous case of determinants and minors. Motivated by the study of the singular locus of the permanental hypersurface, we focus on the codimension of these varieties. We introduce a ${\mathbb{C}}^{\ast }$-action on matrices and prove a number of results. In particular, we improve a lower bound on the codimension of the aforementioned singular locus established by von zur Gathen in 1987.