Nonemptiness of Brill-Noether loci in \(M(2,L)\) (Q2788776)

For a complex, smooth, projective curve \(C\) of genus \(g \geq 2\), let \(M(2,d)\) (respectively \(M(2,L)\)) be the moduli space of rank \(2\) stable vector bundles of degree \(d\) (respectively of determinant \(L\), where \(L\) is a line bundle of degree \(d\)). One denotes by \(B(2,d,k)\) (respectively \(B(2,L,k)\)) the (Brill-Noether) locus of those vector bundles \(E\) in \(M(2,d)\) (respectively in \(M(2,L)\)) with \(h^0(E) \geq k\). In the Introduction, known results (and the methods to prove them) on the nonemptiness of the Brill-Noether locus are discussed. In the paper under review, the case \(d\) odd is studied. One shows that \(B(2,L,k)\) has a virtual fundamental class which is not zero, under certain numerical conditions. Consequently, in those cases, the Brill-Noether locus is not empty. ``For many values of \(d\) and \(k\) the result is best possible.'' The results are improved for \(k \leq 5\). The proof uses Porteous formula and a combinatorial lemma which is proved in the Appendix.