Abstract
Trigonal curves provide an example of Brill–Noether special curves. Theorem 1.3 of Larson (Invent Math 224(3):767–790, 2021) characterizes the Brill–Noether theory of general trigonal curves and the refined stratification by Brill–Noether splitting loci, which parametrize line bundles whose push-forward to \(\mathbb {P}^1\) has a specified splitting type. This note describes the refined stratification for all trigonal curves. Given the Maroni invariant of a trigonal curve, we determine the dimensions of all Brill–Noether splitting loci and describe their irreducible components. When the dimension is positive, these loci are connected, and if furthermore the Maroni invariant is 0 or 1, they are irreducible.
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Acknowledgements
Many key ideas of this paper were discovered with Izzet Coskun during a visit at University of Illinois–Chicago. I am grateful for his insights and interest in discussing this problem. I would also like to thank Ravi Vakil for helpful conversations; and David Eisenbud for asking if [9, Example 1.1] held for all genus 5 trigonal curves, which was the initial motivation for this work. I am also grateful to the Hertz Foundation Graduate Fellowship, NSF Graduate Research Fellowship (under DGE–1656518), Maryam Mirzakhani Graduate Fellowship, and the Stanford Graduate Fellowship for their generous support.
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Larson, H.K. Refined Brill–Noether theory for all trigonal curves. European Journal of Mathematics 7, 1524–1536 (2021). https://doi.org/10.1007/s40879-021-00493-6
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DOI: https://doi.org/10.1007/s40879-021-00493-6