Automorphism loci for the moduli space of rational maps

2014/08/25 by Miasnikov, Nikita, Stout, Brian, Williams, Phillip
#Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1408.5655

Abstract

Let k be an algebraically closed field of characteristic 0 and M_d the moduli space of rational maps on ℙ¹ of degree d over k. This paper describes the automorphism loci A⊂ Rat_d and A⊂ M_d and the singular locus S\subsetM_d. In particular, we determine which groups occur as subgroups of the automorphism group of some [ϕ]\inM_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 M_d, M^s_d, and M^ss_d.

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• Moduli spaces for families of rational maps on P¹
• Indeterminacy loci of iterate maps in moduli space
• The automorphisms group of M_0,n
• On Real and Pseudo-Real Rational Maps