Brill–Noether loci for divisors on irregular varieties

  • Margarida Mendes Lopes

    Instituto Superior Técnico, Lisboa, Portugal
  • Rita Pardini

    Università di Pisa, Italy
  • Gian Pietro Pirola

    Università di Pavia, Italy


We take up the study of the Brill-Noether loci Wr(L,X):={ηPic0(X)  h0(Lη)r+1}W^r(L,X):=\{\eta\in \mathrm {Pic}^0(X)\ |\ h^0(L\otimes\eta)\ge r+1\}, where XX is a smooth projective variety of dimension >1>1, LPic(X)L\in \mathrm {Pic}(X), and r0r\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 h0(KD)h^0(K_D), where DD is a divisor that moves linearly on a smooth projective variety XX of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension >2>2.

In the 22-dimensional case we prove an existence theorem: we define a Brill-Noether number ρ(C,r)\rho(C, r) for a curve CC on a smooth surface XX of maximal Albanese dimension and we prove, under some mild additional assumptions, that if ρ(C,r)0\rho(C, r)\ge 0 then Wr(C,X)W^r(C,X) is nonempty of dimension ρ(C,r)\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.

Cite this article

Margarida Mendes Lopes, Rita Pardini, Gian Pietro Pirola, Brill–Noether loci for divisors on irregular varieties. J. Eur. Math. Soc. 16 (2014), no. 10, pp. 2033–2057

DOI 10.4171/JEMS/482