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A refined Brill–Noether theory over Hurwitz spaces

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Let \(f:C \rightarrow \mathbb {P}^1\) be a degree k genus g cover. The stratification of line bundles \(L \in {{\,\mathrm{Pic}\,}}^d(C)\) by the splitting type of \(f_*L\) is a refinement of the stratification by Brill–Noether loci \(W^r_d(C)\). We prove that for general degree k covers, these strata are smooth of the expected dimension. In particular, this determines the dimensions of all irreducible components of \(W^r_d(C)\) for a general k-gonal curve (there are often components of different dimensions), extending results of Pflueger (Adv Math 312:46–63, 2017) and Jensen and Ranganathan (Brill–Noether theory for curves of a fixed gonality, arXiv:1701.06579, 2017). The results here apply over any algebraically closed field.

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Acknowledgements

This work was inspired by Geoffrey Smith, who introduced the notion of Brill–Noether splitting loci in a seminar at Stanford and asked if the author’s results in [17] could be applied to show their existence. I am grateful for his insight. Thanks also to Ravi Vakil, Eric Larson, Sam Payne, and Melanie Wood for fruitful discussions. I thank Kaelin Cook-Powell and Dave Jensen for their generosity and openness in sharing their work. I am grateful to the Hertz Foundation Graduate Fellowship, NSF Graduate Research Fellowship under grant DGE–1656518, Maryam Mirzakhani Graduate Fellowship, and the Stanford Graduate Fellowship for their generous support.

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    Hannah K. Larson

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Larson, H.K. A refined Brill–Noether theory over Hurwitz spaces. Invent. math. 224, 767–790 (2021). https://doi.org/10.1007/s00222-020-01023-z

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