## 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.

### MSC:

 14C20 Divisors, linear systems, invertible sheaves 14J29 Surfaces of general type 14H51 Special divisors on curves (gonality, Brill-Noether theory)

### Keywords:

irregular variety; Brill-Noether theory; Albanese dimension
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### 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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