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

Hidalgo, Rubén A.; Quispe, Saúl

doi:10.1017/s0017089516000665

Cited by 2 publications

(3 citation statements)

References 10 publications

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“…The above, in particular, permits to observe that B 2 = ∅ is connected. For d 3, it is known that B d = ∅ is the locus where M d fails to be a topological manifold [14] and, in [11], it was observed that B d is always connected.…”

Section: The Branch Locus Of Moduli Space the Branch Locus Of Mmentioning

confidence: 99%

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On real and pseudo-real rational maps ^*

Hidalgo

Quispe

2021

Nonlinearity

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The moduli space M d , of complex rational maps of degree d ⩾ 2, is a connected complex orbifold which carries a natural real structure, coming from usual complex conjugation. Its real points are the classes of rational maps admitting antiholomorphic automorphisms. The locus of the real points M d ( R ) decomposes as a disjoint union of the loci M d R , consisting of the real rational maps, and P d , consisting of the pseudo-real ones. We obtain that, both M d R and M d ( R ) , are connected and that P d is disconnected. We also observe that the group of holomorphic automorphisms of a pseudo-real rational map is either trivial or a cyclic group. For every n ⩾ 1, we construct pseudo-real rational maps whose group of holomorphic automorphisms is cyclic of order n. As the field of moduli of a pseudo-real rational map is contained in R , these maps provide examples of rational maps which are not definable over their field of moduli. It seems that these are the only explicit examples in the literature (Silverman) of rational maps which cannot be defined over their field of moduli. We provide explicit examples of real rational maps which cannot be defined over their field of moduli. Finally, we also observe that every real rational map, which admits a model over the algebraic numbers, can be defined over the real algebraic numbers.

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Section: The Branch Locus Of Moduli Space the Branch Locus Of Mmentioning

confidence: 99%

“…In this case, dim C (M d,Z 2 ) = d − 1, in particular, M d,Z 2 = ∅. In [11], we used this observation to obtain the connectedness of B d . Remark 5.…”

Section: Corollary 1 ([14]mentioning

confidence: 99%

On real and pseudo-real rational maps ^*

Hidalgo

Quispe

2021

Nonlinearity

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“…where the cuspid (−6, 12) corresponds to the class of a rational map φ(z) = 1/z 2 with Aut(φ) D 3 (the dihedral group of order 6); all other points in the cubic correspond to those rational maps with the cyclic group Z 2 as full group of holomorphic automorphisms. In [5] it was observed that B d is always connected.…”

Section: Introductionmentioning

confidence: 99%

On Real and Pseudo-Real Rational Maps

Hidalgo¹,

Quispe²

2015

Preprint

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Moduli space of rational maps carries a natural real structure, coming from usual complex conjugation. Its real points are the classes of rational maps admitting an antiholomorphic automorphism. We provide a characterization of those rational maps admitting antiholomorphic automorphisms and observe that the real locus is connected. These maps are either real or pseudo-real rational maps.Pseudo-real rational maps never can be defined over their field of moduli. Their groups of holomorphic automorphisms is either trivial or a cyclic group. A characterization, in terms of the corresponding group of holomorphic automorphisms, is provided and we show that, for every n ≥ 1, there is one whose group of holomorphic automorphisms has order n. As a consequence, we obtain that the locus in moduli space of equivalence classes of pseudo-real rational maps is non-connected.We provide examples of real rational maps which cannot be defined over their field of moduli and observe that every real rational map, which admits a model over the algebraic numbers, must also admit another model over the real algebraic numbers.

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

Cited by 2 publications

References 10 publications

On real and pseudo-real rational maps ^*

On real and pseudo-real rational maps ^*

On Real and Pseudo-Real Rational Maps

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

Cited by 2 publications

References 10 publications

On real and pseudo-real rational maps *

On real and pseudo-real rational maps *

On Real and Pseudo-Real Rational Maps

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Product

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On real and pseudo-real rational maps ^*

On real and pseudo-real rational maps ^*