New examples of twisted Brill-Noether loci. I (Q6573199)

Let \(C\) be a smooth irreducible complex projective curve of genus \(g \geq 2\), and \(M_i=M(n_i,d_i)\), \(i=1,2\), the moduli space of stable bundles of rank \(n_i\) and degree \(d_i\). The twisted Brill-Noether locus \(B(n_1,d_1,k)(E_2)\), \(k \in {\mathbb N}\), \(E_2\in M_2\) is defined as the vector bundles \(E_1\in M_1\) such that \(h^0(E_1 \otimes E_2) \geq k\). This locus is a closed subscheme of \(M_1\) and a natural \textit{expected dimension} appears. If gcd\((n_i,d_i)=1\) then there exist universal bundles \({\mathcal U}_i\) on \(M_i \times C\) and the universal twisted Brill-Noether locus \(B^k({\mathcal U}_1,{\mathcal U}_2)\) is defined as the pairs \((E_1,E_2) \in M_1 \times M_2\) such that \(h^0(E_1 \otimes E_2) \geq k\). The definition extends to the non-coprime case, and \(B^k({\mathcal U}_1,{\mathcal U}_2) \subset M_1 \times M_2\) is also a closed subscheme with a notion of expected dimension. \N\NThe first main theorem of the paper under review (see Thm. 1.2) proves the non-emptiness of \(B^k({\mathcal U}_1,{\mathcal U}_2)\), \(k=k_1k_2\) under the assumption of the non-emptiness of \(B(n_i,d_i,k_i)({\mathcal O}_C)\) and some numerical hypotheses on \(n_i,d_i,g\). Some of this non-empty Brill-Noether loci have negative expected dimension. This result is applied (see Thm. 4.8) to show a new region in the Brill-Noether map. The third main theorem (see Thm. 1.3) also proves the non-emptiness of certain Brill-Noether loci, some of them of negative expected dimension. In this last result \(d_2<0\), so that \(h^0(E_2)=0\) for any \(E_2 \in M_2\).