2011/12/29 by Margarida Mendes Lopes, Rita Pardini, Lopes, Margarida Mendes +3
Mathematics · #14C20 #14H51 #14J29 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14C20 #msc:14H51 #msc:14J29
paper · pdf · doi:10.48550/arxiv.1112.6357
Final version, to appear in Journal of the European Mathematical Society
arxiv created 2012/10/07 · arxiv updated 2012/10/09
For a projective variety X, a line bundle L on X and r a natural number we consider the r-th Brill-Noether locus Wr(L,X):=η∈ Pic0(X)|h0(L+η)≥ 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 ρ(C, r) and we prove, under some mild additional assumptions, that if ρ(C, r) is non negative then Wr(C,X) is nonempty of dimension bigger or equal to ρ(C,r). As an application, we derive lower bounds for h0(KD) 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.