#### Hall 123

#### Non-emptiness of Brill-Noether loci over certain surfaces.

#### Sarbeswar Pal

##### IISER, Thiruvananthapuram

*Brill-Noether loci of moduli space of stable bundles over arbitrary irreducible, smooth, projective*

Variety (over complex numbers) was constructed by L. Costa and R. M. Miro-Roig. This is a generalization of classical Brill-Noether loci over curves. In this lecture, we will outline this construction, and define "Petri map"over surfaces, as an analouge of the case of curves. We will prove the non-emptiness of certain Brill-Noether loci of moduli space of stable bundles over very general smooth quintic hypersurface in P^3, and use the Petri map to prove the existence of smooth point in the Brill-Noether loci.

Done