Brill-Noether loci for divisors on irregular varieties

TL;DR

This paper studies Brill-Noether loci for divisors on irregular varieties, describing their scheme structure, tangent spaces, and establishing conditions for nonemptiness and dimension bounds, with applications to divisors and curves on surfaces of maximal Albanese dimension.

Contribution

It introduces a scheme-theoretic description of Brill-Noether loci on irregular varieties and proves nonemptiness and dimension bounds under certain conditions, extending classical Brill-Noether theory.

Findings

W^r(L,X) scheme structure described

Nonemptiness of W^r(C,X) when ho(C,r) ≥ 0

Lower bounds for h^0(K_D) and inequalities for curve invariants

Abstract

For a projective variety X, a line bundle L on X and r a natural number we consider the r-th Brill-Noether locus W^r(L,X):={\eta\in Pic^0(X)|h^0(L+\eta)\geq r+1}: we describe its natural scheme structure and compute the Zariski tangent space. If X is a smooth surface of maximal Albanese dimension and C is a curve on X, we define a Brill-Noether number \rho(C, r) and we prove, under some mild additional assumptions, that if \rho(C, r) is non negative then W^r(C,X) is nonempty of dimension bigger or equal to \rho(C,r). As an application, we derive lower bounds for h^0(K_D) for a divisor D that moves linearly on a smooth projective variety X of maximal Albanese dimension and inequalities for the numerical invariants of curves that do not move linearly on a surface of maximal Albanese dimension.