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Automorphism loci for the moduli space of rational maps

Volume 180 / 2017

Nikita Miasnikov, Brian Stout, Phillip Williams Acta Arithmetica 180 (2017), 267-296 MSC: Primary 37P45; Secondary 14D22. DOI: 10.4064/aa8548-6-2017 Published online: 30 August 2017

Abstract

Let $k$ be an algebraically closed field of characteristic $0$, and $\mathcal{M}_d$ the moduli space of rational maps on $\mathbb P^1$ of degree $d$ over $k$. This paper describes the automorphism loci $A\subset \mathrm{Rat}_d$ and $\mathcal{A}\subset \mathcal{M}_d$ and the singular locus $\mathcal{S}\subset\mathcal{M}_d$. In particular, we determine which groups occur as subgroups of the automorphism group of some $[\phi]\in\mathcal{M}_d$ for a given $d$ and calculate the dimension of the locus. Next, we prove an analogous theorem to the Rauch–Popp–Oort characterization of singular points on the moduli scheme for curves. The results concerning these distinguished loci are used to compute the Picard and class groups of $\mathcal{M}_d$, $\mathcal{M}_d^s$, and $\mathcal{M}_d^{ss}$.

Authors

  • Nikita MiasnikovDepartment of Mathematics
    SUNY Oswego
    7060 Route 104
    Oswego, NY 13126, U.S.A.
    e-mail
  • Brian StoutMinerva Schools at KGI
    1145 Market St.
    San Francisco, CA 94103, U.S.A.
    e-mail
  • Phillip WilliamsDepartment of Mathematics
    The King’s College
    56 Broadway
    New York, NY 10004, U.S.A.
    e-mail

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