### Margarida Mendes Lopes

Instituto Superior Técnico, Lisboa, Portugal### Rita Pardini

Università di Pisa, Italy### Gian Pietro Pirola

Università di Pavia, Italy

We take up the study of the Brill-Noether loci $W_{r}(L,X):={η∈Pic_{0}(X)∣h_{0}(L⊗η)≥r+1}$, where $X$ is a smooth projective variety of dimension $>1$, $L∈Pic(X)$, and $r≥0$ 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 $h_{0}(K_{D})$, where $D$ is a divisor that moves linearly on a smooth projective variety $X$ of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension $>2$.

In the $2$-dimensional case we prove an existence theorem: we define a Brill-Noether number $ρ(C,r)$ for a curve $C$ on a smooth surface $X$ of maximal Albanese dimension and we prove, under some mild additional assumptions, that if $ρ(C,r)≥0$ then $W_{r}(C,X)$ is nonempty of dimension $≥ρ(C,r)$.

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.

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. 2033–2057

DOI 10.4171/JEMS/482