Singular locus of the theta divisor for vector bundles of rank two (Q1897998)

Let \(F\) be a generic vector bundle of rank \(n'\) and degree \(d'\). Denote by \(W^r_{n,d} (F)\) the sets of vector bundles \(E\) of rank \(n\) and degree \(d\) such that \(h^0 (E \otimes F) \geq r + 1\). When \(n = 1\) and \(F\) is a line bundle, these are the classical Brill-Noether loci. From the theory of determinantal varieties, one expects the singular locus to be \(W^{r + 1}_{n,d} (F)\). This is true in the classical case. We show that the result fails for \(n = 2\) and rank \(F = 2\) and we determine exactly the singular locus in this case.