Brill-Noether loci for divisors on irregular varieties

WetakeupthestudyoftheBrill–NoetherlociWr(L,X):={η∈Pic0(X)|h0(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 h0(KD), 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 Wr(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.