Brill–Noether loci for divisors on irregular varieties

Abstract

We take up the study of the Brill-Noether loci $W^r(L,X):=\{\eta \in {\mathrm{Pic}}^0(X)|h^0(L\otimes \eta )\ge r+1\}$, where $X$ is a smooth projective variety of dimension $>1$, $L\in \mathrm{Pic}(X)$, and $r\ge 0$ 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 $\rho (C,r)$ for a curve $C$ on a smooth surface $X$ of maximal Albanese dimension and we prove, under some mild additional assumptions, that if $\rho (C,r)\ge 0$ then $W^r(C,X)$ is nonempty of dimension $\ge \rho (C,r)$.

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.