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Online ISSN 1534-7486; Print ISSN 1056-3911
The Kodaira classification of the moduli of hyperelliptic curves
Authors:
Ignacio Barros and Scott Mullane; with an appendix by Irene Schwarz
Journal:
J. Algebraic Geom.
DOI:
https://doi.org/10.1090/jag/843
Published electronically:
April 15, 2025
Full-text PDF
Abstract | References | Additional Information
Abstract: We study the birational geometry of the moduli spaces of hyperelliptic curves with marked points. We show that these moduli spaces have non-$\mathbb {Q}$-factorial singularities. We complete the Kodaira classification by proving that these spaces have Kodaira dimension $4g+3$ when the number of markings is $4g+6$ and are of general type when the number of markings is $n\geq 4g+7$. Similarly, we consider the natural finite cover given by ordering the Weierstrass points. In this case, we provide a full Kodaira classification showing that the Kodaira dimension is negative when $n\leq 3$, one when $n=4$, and of general type when $n\geq 5$. For this, we carry out a singularity analysis of ordered and unordered pointed Hurwitz spaces. We show that the ordered space has canonical singularities and the unordered space has noncanonical singularities. We describe all noncanonical points and show that pluricanonical forms defined on the full regular locus extend to any resolution. Further, we provide a full classification of the structure of the pseudo-effective cone of Cartier divisors for the moduli space of hyperelliptic curves with marked points. We show the cone is nonpolyhedral when the number of markings is at least two and polyhedral in the remaining cases.
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Additional Information
Ignacio Barros
Affiliation:
Department of Mathematics, Universiteit Antwerpen, 2020 Antwerpen, Belgium
MR Author ID:
1276538
ORCID:
0000-0002-7729-9413
Email:
ignacio.barros@uantwerpen.be
Scott Mullane
Affiliation:
School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia
MR Author ID:
1215575
Email:
mullanes@unimelb.edu.au
Irene Schwarz
Affiliation:
Departement Mathematik, ETH Zürich, 8092 Zürich, Switzerland
MR Author ID:
1217299
Received by editor(s):
October 3, 2022
Received by editor(s) in revised form:
October 1, 2023
Published electronically:
April 15, 2025
Additional Notes:
The first author was supported by the ERC Synergy Grant ERC-2020-SyG-854361-HyperK, the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – SFB-TRR 358/1 2023 – 491392403, and the Research Foundation Flanders (FWO) within the framework of the Odysseus program project number G0D9323N. The second author was supported by the Alexander von Humboldt Foundation, ERC Advanced Grant SYZYGY, and DECRA Grant DE220100918 from the Australian Research Council.
Article copyright:
© Copyright 2025
University Press, Inc.