Brill-Noether loci for divisors on irregular varieties
Abstract
For a projective variety X, a line bundle L on X and r a natural number we consider the r-th Brill-Noether locus W^r(L,X):={\eta\in Pic^0(X)|h^0(L+\eta)\geq r+1}: we describe its natural scheme structure and compute the Zariski tangent space. If X is a smooth surface of maximal Albanese dimension and C is a curve on X, we define a Brill-Noether number \rho(C, r) and we prove, under some mild additional assumptions, that if \rho(C, r) is non negative then W^r(C,X) is nonempty of dimension bigger or equal to \rho(C,r). As an application, we derive lower bounds for h^0(K_D) for a divisor D that moves linearly on a smooth projective variety X of maximal Albanese dimension and inequalities for the numerical invariants of curves that do not move linearly on a surface of maximal Albanese dimension.
Cite
@article{arxiv.1112.6357,
title = {Brill-Noether loci for divisors on irregular varieties},
author = {Margarida Mendes Lopes and Rita Pardini and Gian Pietro Pirola},
journal= {arXiv preprint arXiv:1112.6357},
year = {2012}
}
Comments
Final version, to appear in Journal of the European Mathematical Society