We take up the study of the Brill-Noether loci , where is a smooth projective variety of dimension , , and is an integer.
By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for , where is a divisor that moves linearly on a smooth projective variety of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension .
In the -dimensional case we prove an existence theorem: we define a Brill-Noether number for a curve on a smooth surface of maximal Albanese dimension and we prove, under some mild additional assumptions, that if then is nonempty of dimension .
Inequalities for the numerical invariants of curves that do not move linearly on a surface of maximal Albanese dimension are obtained as an application of the previous results.
Cite this article
Margarida Mendes Lopes, Rita Pardini, Gian Pietro Pirola, Brill–Noether loci for divisors on irregular varieties. J. Eur. Math. Soc. 16 (2014), no. 10, pp. 2033–2057DOI 10.4171/JEMS/482