Enumerating pencils with moving ramification on curves
Author:
Carl Lian
Journal:
J. Algebraic Geom. 32 (2023), 143-182
DOI:
https://doi.org/10.1090/jag/776
Published electronically:
December 28, 2021
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Abstract |
References |
Additional Information
Abstract: We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve $E$, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps $E\to \mathbb {P}^1$ with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.
References
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- Dan Edidin, Brill-Noether theory in codimension-two, J. Algebraic Geom. 2 (1993), no. 1, 25–67. MR 1185606
- David Eisenbud and Joe Harris, Limit linear series: basic theory, Invent. Math. 85 (1986), no. 2, 337–371. MR 846932, DOI 10.1007/BF01389094
- David Eisenbud and Joe Harris, Irreducibility of some families of linear series with Brill-Noether number $-1$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 33–53. MR 985853, DOI 10.24033/asens.1574
- Gavril Farkas, Brill-Noether with ramification at unassigned points, J. Pure Appl. Algebra 217 (2013), no. 10, 1838–1843. MR 3053519, DOI 10.1016/j.jpaa.2013.01.016
- Gavril Farkas, Riccardo Moschetti, Juan Carlos Naranjo, and Gian Pietro Pirola, Alternating Catalan numbers and curves with triple ramification, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (2021), no. 2, 665–690. MR 4288668
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- J. Harris, On the Kodaira dimension of the moduli space of curves. II. The even-genus case, Invent. Math. 75 (1984), no. 3, 437–466. MR 735335, DOI 10.1007/BF01388638
- Fu Liu and Brian Osserman, The irreducibility of certain pure-cycle Hurwitz spaces, Amer. J. Math. 130 (2008), no. 6, 1687–1708. MR 2464030, DOI 10.1353/ajm.0.0031
- Adam Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125 (2003), no. 1, 105–138. MR 1953519, DOI 10.1353/ajm.2003.0005
- E. Mukhin, V. Tarasov, and A. Varchenko, Schubert calculus and representations of the general linear group, J. Amer. Math. Soc. 22 (2009), no. 4, 909–940. MR 2525775, DOI 10.1090/S0894-0347-09-00640-7
- Brian Osserman, The number of linear series on curves with given ramification, Int. Math. Res. Not. 47 (2003), 2513–2527. MR 2007538, DOI 10.1155/S1073792803131492
- Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
References
- Dan Abramovich, Alessio Corti, and Angelo Vistoli, Twisted bundles and admissible covers, Comm. Algebra 31 (2003), no. 8, 3547–3618. Special issue in honor of Steven L. Kleiman. MR 2007376, DOI 10.1081/AGB-120022434
- Dan Edidin, Brill-Noether theory in codimension-two, J. Algebraic Geom. 2 (1993), no. 1, 25–67. MR 1185606
- David Eisenbud and Joe Harris, Limit linear series: basic theory, Invent. Math. 85 (1986), no. 2, 337–371. MR 846932, DOI 10.1007/BF01389094
- David Eisenbud and Joe Harris, Irreducibility of some families of linear series with Brill-Noether number $-1$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 33–53. MR 985853
- Gavril Farkas, Brill-Noether with ramification at unassigned points, J. Pure Appl. Algebra 217 (2013), no. 10, 1838–1843. MR 3053519, DOI 10.1016/j.jpaa.2013.01.016
- Gavril Farkas, Riccardo Moschetti, Juan Carlos Naranjo, and Gian Pietro Pirola, Alternating Catalan numbers and curves with triple ramification, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (2021), no. 2, 665–690. MR 4288668
- Gavril Farkas and Nicola Tarasca, Pointed Castelnuovo numbers, Math. Res. Lett. 23 (2016), no. 2, 389–404. MR 3512891, DOI 10.4310/MRL.2016.v23.n2.a5
- J. Harris, On the Kodaira dimension of the moduli space of curves. II. The even-genus case, Invent. Math. 75 (1984), no. 3, 437–466. MR 735335, DOI 10.1007/BF01388638
- Fu Liu and Brian Osserman, The irreducibility of certain pure-cycle Hurwitz spaces, Amer. J. Math. 130 (2008), no. 6, 1687–1708. MR 2464030, DOI 10.1353/ajm.0.0031
- Adam Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125 (2003), no. 1, 105–138. MR 1953519
- E. Mukhin, V. Tarasov, and A. Varchenko, Schubert calculus and representations of the general linear group, J. Amer. Math. Soc. 22 (2009), no. 4, 909–940. MR 2525775, DOI 10.1090/S0894-0347-09-00640-7
- Brian Osserman, The number of linear series on curves with given ramification, Int. Math. Res. Not. 47 (2003), 2513–2527. MR 2007538, DOI 10.1155/S1073792803131492
- Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
Additional Information
Carl Lian
Affiliation:
Institut für Mathematik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
MR Author ID:
1096109
Email:
liancarl@hu-berlin.de
Received by editor(s):
October 6, 2020
Received by editor(s) in revised form:
November 10, 2020
Published electronically:
December 28, 2021
Additional Notes:
This project was undertaken with the support of an NSF Graduate Research Fellowship
Article copyright:
© Copyright 2021
University Press, Inc.
