##
**Brill-Noether loci for divisors on irregular varieties.**
*(English)*
Zbl 1317.14019

The authors define, given a projective variety \(X\), a line bundle \(L\) on \(X\) and an integer \(r\geq 0\), the locus
\[
W^r(L,X) := \{\eta \in \mathrm{Pic}^0(X)|h^0(L \otimes \eta) \geq r + 1\}
\]
analog of the classical Brill-Noether loci on curves.

They mainly focus on two cases: singular projective curves (say \(C\)) and smooth surfaces (say \(S\)) of maximal Albanese dimension.

In the first case \(\mathrm{Pic}^0(X)\) is not necessarily an abelian variety. Then they study the intersection of \(W^r(L,X)\) with a compact subgroup \(T\) of dimension \(t\), and show that (under a technical assumption on \(T\)) this intersection has dimension at least equal to the Brill-Noether number \(\rho(t, r, d) = t -(r +1)(p_a(C)-d +r)\) (here \(d\) is the total degree of \(L\); in particular the intersection is not empty if \(\rho \geq 0\)).

The second case uses extensively the results of the first case, as one naturally associates to every effective divisor \(C \subset S\) a restriction map \(\mathrm{Pic}^0(S) \rightarrow \mathrm{Pic}^0(C)\) and the image is a compact subgroup. Then for \(L = {\mathcal O}_C(C)\) the Brill-Noether number equals \(\rho(C, r) := q - (r + 1)(p_a(C) - C^2 + r)\): here \(q\) is the irregularity of \(S\). Then the authors prove that \(W^r(C, S)\) is nonempty of dimension at least \(\min(q, \rho(C, r))\) under the assumption that \(\rho(C, r) > 1\) (they prove it also for \(\rho = 1\) under an additional hypothesis).

A second theme of the paper is the study of the restriction map \(r_D : H^0(K_X) \rightarrow H0(K_{X|D})\) when \(X\) is a smooth variety of maximal Albanese dimension and \(D \subset X\) is an effective divisor whose Albanese image generates the Albanese variety of \(X\): the authors obtain some lower bounds for the rank of this map (under some assumptions). These results, together with the results above on the Brill-Noether locus are then used to study curves on a surfaces of general type with \(q > 1\) and not fibred onto a curve of genus \(> 1\), in particular giving inequalities for the numerical invariants of a rigid curve whose irreducible components have positive selfintersection.

Finally they find a connection among the fixed part of the main paracanonical system and the ramification locus of the Albanese map. More precisely they show that if \(S\) is a smooth surface of general type with irregularity \(q \geq 2\) and no irrational pencils of genus \(> q/2\), every irreducible curve with positive selfintersection contained in the fixed part of the main paracanonical system is also contained in the ramification locus of the Albanese map.

They mainly focus on two cases: singular projective curves (say \(C\)) and smooth surfaces (say \(S\)) of maximal Albanese dimension.

In the first case \(\mathrm{Pic}^0(X)\) is not necessarily an abelian variety. Then they study the intersection of \(W^r(L,X)\) with a compact subgroup \(T\) of dimension \(t\), and show that (under a technical assumption on \(T\)) this intersection has dimension at least equal to the Brill-Noether number \(\rho(t, r, d) = t -(r +1)(p_a(C)-d +r)\) (here \(d\) is the total degree of \(L\); in particular the intersection is not empty if \(\rho \geq 0\)).

The second case uses extensively the results of the first case, as one naturally associates to every effective divisor \(C \subset S\) a restriction map \(\mathrm{Pic}^0(S) \rightarrow \mathrm{Pic}^0(C)\) and the image is a compact subgroup. Then for \(L = {\mathcal O}_C(C)\) the Brill-Noether number equals \(\rho(C, r) := q - (r + 1)(p_a(C) - C^2 + r)\): here \(q\) is the irregularity of \(S\). Then the authors prove that \(W^r(C, S)\) is nonempty of dimension at least \(\min(q, \rho(C, r))\) under the assumption that \(\rho(C, r) > 1\) (they prove it also for \(\rho = 1\) under an additional hypothesis).

A second theme of the paper is the study of the restriction map \(r_D : H^0(K_X) \rightarrow H0(K_{X|D})\) when \(X\) is a smooth variety of maximal Albanese dimension and \(D \subset X\) is an effective divisor whose Albanese image generates the Albanese variety of \(X\): the authors obtain some lower bounds for the rank of this map (under some assumptions). These results, together with the results above on the Brill-Noether locus are then used to study curves on a surfaces of general type with \(q > 1\) and not fibred onto a curve of genus \(> 1\), in particular giving inequalities for the numerical invariants of a rigid curve whose irreducible components have positive selfintersection.

Finally they find a connection among the fixed part of the main paracanonical system and the ramification locus of the Albanese map. More precisely they show that if \(S\) is a smooth surface of general type with irregularity \(q \geq 2\) and no irrational pencils of genus \(> q/2\), every irreducible curve with positive selfintersection contained in the fixed part of the main paracanonical system is also contained in the ramification locus of the Albanese map.

Reviewer: Roberto Pignatelli (Trento)

### MSC:

14C20 | Divisors, linear systems, invertible sheaves |

14J29 | Surfaces of general type |

14H51 | Special divisors on curves (gonality, Brill-Noether theory) |

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\textit{M. Mendes Lopes} et al., J. Eur. Math. Soc. (JEMS) 16, No. 10, 2033--2057 (2014; Zbl 1317.14019)

### References:

[1] | Alexeev, V.: Compactified Jacobians and Torelli map. Publ. RIMS Kyoto Univ. 40, 1241-1265 (2004) · Zbl 1079.14019 · doi:10.2977/prims/1145475446 |

[2] | Arbarello, E., Cornalba, M., Griffiths, P. A.: Geometry of Algebraic Curves. Vol. II. Grundlehren Math. Wiss. 268, Springer, Heidelberg (2011) · Zbl 1235.14002 · doi:10.1007/978-3-540-69392-5 |

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