The singular locus of the variety of matrices with rank at most r is the variety of matrices with rank at most (r-1). This fact is independent of the size of the matrices provided it is strictly bigger than r.

A natural environment for varieties stable under the action of the general linear group and defined uniformly with respect to the dimension is the setting of polynomial functors, where we name these Vec-varieties.

In this setting, the phenomenon above happens to be more than a coincidence. Indeed, in a joint work with Christopher H. Chiu and Jan Draisma, we proved that each Vec-variety admits a unique Vec-subvariety coinciding with the singular locus when the dimension is big enough.

The aim of this talk is to provide the necessary background, and a proof of the result.