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Limit linear series and ranks of multiplication maps

Fu Liu Department of Mathematics, University of California, Davis CA 95616 Brian Osserman Department of Mathematics, University of California, Davis CA 95616 Montserrat Teixidor i Bigas Department of Mathematics, Tufts University, Medford MA 02155 Naizhen Zhang Institute of Differential Geometry, Universität Hannover, 30167 Hannover, GERMANY

Algebraic Geometry mathscidoc:2203.45010

Transactions of the American Mathematical Society, 374, 367-405, 2020.11
We develop a new technique to study ranks of multiplication maps for linear series via limit linear series and degenerations to chains of elliptic curves. We prove an elementary criterion and apply it to proving cases of the Maximal Rank Conjecture. We give a new proof of the case of quadrics, and also treat several families in the case of cubics. Our proofs do not require restrictions on direction of approach, so we recover new information on the locus in the moduli space of curves on which the maximal rank condition fails.
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@inproceedings{fu2020limit,
  title={Limit linear series and ranks of multiplication maps},
  author={Fu Liu, Brian Osserman, Montserrat Teixidor i Bigas, and Naizhen Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220317170202656879003},
  booktitle={Transactions of the American Mathematical Society},
  volume={374},
  pages={367-405},
  year={2020},
}
Fu Liu, Brian Osserman, Montserrat Teixidor i Bigas, and Naizhen Zhang. Limit linear series and ranks of multiplication maps. 2020. Vol. 374. In Transactions of the American Mathematical Society. pp.367-405. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220317170202656879003.
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