Automorphism loci for the moduli space of rational maps

Nikita Miasnikov¹ ; Brian Stout² ; Phillip Williams³

¹ Department of Mathematics SUNY Oswego 7060 Route 104 Oswego, NY 13126, U.S.A.
² Minerva Schools at KGI 1145 Market St. San Francisco, CA 94103, U.S.A.
³ Department of Mathematics The King’s College 56 Broadway New York, NY 10004, U.S.A.

Acta Arithmetica, Tome 180 (2017) no. 3, pp. 267-296

Résumé

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 $\mathcal{M}_d^{ss}$.

Détail

DOI : 10.4064/aa8548-6-2017

Keywords: algebraically closed field characteristic nbsp mathcal moduli space rational maps mathbb degree paper describes automorphism loci subset mathrm rat mathcal subset mathcal singular locus mathcal subset mathcal particular determine which groups occur subgroups automorphism group phi mathcal given calculate dimension locus prove analogous theorem rauch popp oort characterization singular points moduli scheme curves results concerning these distinguished loci compute picard class groups mathcal mathcal nbsp mathcal