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.