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Abel maps and limit linear series

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Abstract

We explore the relationship between limit linear series and fibers of Abel maps in the case of curves with two smooth components glued at a single node. To an \(r\)-dimensional limit linear series satisfying a certain exactness property (weaker than the refinedness property of Eisenbud and Harris) we associate a closed subscheme of the appropriate fiber of the Abel map. We then describe this closed subscheme explicitly, computing its Hilbert polynomial and showing that it is Cohen–Macaulay of pure dimension \(r\). We show that this construction is also compatible with one-parameter smoothings.

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Acknowledgments

We would like to thank the referee for a careful reading.

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Authors and Affiliations

  1. Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, Rio de Janeiro, RJ, Brasil

    Eduardo Esteves

  2. Department of Mathematics, University of California, One Shields Avenue, Davis, CA, 95616, USA

    Brian Osserman

Authors
  1. Eduardo Esteves
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  2. Brian Osserman
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Corresponding author

Correspondence to Brian Osserman.

Additional information

The first author was supported by CNPq, Proc. 303797/2007-0 and 473032/2008-2, and FAPERJ, Proc. E-26/102.769/2008 and E-26/110.556/2010. This project was initiated while the authors were visiting MSRI for the 2009 program in Algebraic Geometry and concluded when the second author was visiting IMPA. The authors thank both institutes for the support given.

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Esteves, E., Osserman, B. Abel maps and limit linear series. Rend. Circ. Mat. Palermo 62, 79–95 (2013). https://doi.org/10.1007/s12215-013-0111-0

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  • DOI: https://doi.org/10.1007/s12215-013-0111-0

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