Automorphism loci for the moduli space of rational maps

Miasnikov, Nikita; Stout, Brian; Williams, Phillip E.

doi:10.48550/arxiv.1408.5655

Cited by 3 publications

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“…• Icosahedral group A 5 (or alternating group on 5 elements. Also, from the classification of finite subgroups of PGL 2 , all isomorphic subgroups are in fact conjugate to each other [31,Lemma 2.3]. Thus, given a group, we can simply fix an SL 2 representation and work only with this representation.…”

Section: Automorphism Loci In Mmentioning

confidence: 99%

“…From previous work of Miasnikov, Stout, and Williams [31] (henceforth abbreviated as MSW), we are able to determine the dimension of A 3 (Γ) for any Γ.…”

Section: Automorphism Loci In Mmentioning

confidence: 99%

“…Proof. The groups that occur and the dimensions are calculated in Section 2 of MSW [31]. The second half follows from the observation that if G is a subgroup of H, then A d (H) ⊆ A d (G).…”

Section: Automorphism Loci In Mmentioning

confidence: 99%

“…Such an α is called an automorphism of f , and the set of all such automorphisms forms a group Aut(f ) := {α ∈ PGL 2 : f α = f }. In additional to being special from the existence of these extra symmetries, conjugacy classes with nontrivial automorphisms are exactly the singular points of the moduli space M d for d ≥ 3 [31].…”

Section: Introductionmentioning

confidence: 99%

“…Section 2 gives parameterizations of the automorphism locus A 3 ⊂ M 3 . Taking the classification of finite subgroups of PGL 2 given in Silverman [41] combined with the known dimensions of A 3 (Γ) from Miasnikov-Stout-Williams [31], we find families in the parameter space Rat 3 that map finite-to-one onto families in M 3 of the appropriate dimension. Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

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Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line

Gontmacher,

Hutz,

Jorgenson

et al. 2020

Preprint

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Let f be an endomorphism of the projective line. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group. The group of automorphisms, or stabilizer group, of a given f for this action is known to be a finite group. We determine explicit families that parameterize all endomorphisms defined over Q of degree 3 and degree 4 that have a nontrivial automorphism, the automorphism locus of the moduli space of dynamical systems. We analyze the geometry of these loci in the appropriate moduli space of dynamical systems. Further, for each family of maps, we study the possible structures of Q-rational preperiodic points which occur under specialization.

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Section: Automorphism Loci In Mmentioning

confidence: 99%

“…From previous work of Miasnikov, Stout, and Williams [31] (henceforth abbreviated as MSW), we are able to determine the dimension of A 3 (Γ) for any Γ.…”

Section: Automorphism Loci In Mmentioning

confidence: 99%

Section: Automorphism Loci In Mmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line

Gontmacher,

Hutz,

Jorgenson

et al. 2020

Preprint

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A Note on the Connectedness of the Branch Locus of Rational Maps

Hidalgo

Quispe

2017

Glasgow Math. J.

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Abstract. Milnor proved that the moduli space M d of rational maps of degree d ≥ 2 has a complex orbifold structure of dimension 2(d − 1). Let us denote by S d the singular locus of M d and by B d the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify M 2 with ‫ރ‬ 2 and, within that identification, that B 2 is a cubic curve; so B 2 is connected and S 2 = ∅. If d ≥ 3, then it is well known that S d = B d . In this paper, we use simple arguments to prove the connectivity of S d .

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A classification of degree 2 semi-stable rational maps ℙ 2 →ℙ 2 with large finite dynamical automorphism group

Silverman¹

2019

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Let K be an algebraically closed field of characteristic 0. In this paper we classify the PGL 3 (K)-conjugacy classes of semi-stable dominant degree 2 rational maps f : P 2is finite and of order at least 3. In particular, we prove that #Aut(f ) ≤ 24 in general, that #Aut(f ) ≤ 21 for morphisms, and that #Aut(f ) ≤ 6 for all but finitely many conjugacy classes of f . Abstract. Soit K un corps algébriquement clos de charactérisktique 0. Dans cet article nous classifions les PGL 3 (K)-classes de conjugaison de fonctions rationelles f : P 2 K P 2 K de degré 2 dominantes et semi-stables dont le groupe d'automorphismes Aut(f ) := φ ∈ PGL 3 (K) : φ −1 • f • φ = f est fini et d'ordre au moins 3. En particulier, nous démontrons que #Aut(f ) ≤ 24 en général, que #Aut(f ) ≤ 21 pour les morphismes et que #Aut(f ) ≤ 6 pour toutes excepté un nombre fini de classes de conjugaisons de f .

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Automorphism loci for the moduli space of rational maps

Cited by 3 publications

References 0 publications

Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line

Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line

A Note on the Connectedness of the Branch Locus of Rational Maps

A classification of degree 2 semi-stable rational maps ℙ 2 →ℙ 2 with large finite dynamical automorphism group

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