On explicit descent of marked curves and maps

Sijsling, Jeroen; Voight, John

doi:10.1007/s40993-016-0057-3

“…To show that the Belyȋ map descends, it suffices[8, Cor. 5.4] (or[15, Theorem 3.4.8] with R = ∅) to show that the canonical model E 0 of E corresponding to the cocycle defined by the first two entries of (4.5.4) has a rational point. It does; in fact E 0 is isomorphic to E. Still, none of the points on E 0 that correspond to the ramification points of E are rational over Q(…”

mentioning

confidence: 99%

A database of Belyi maps

Musty

¹

,

Schiavone

²

,

Sijsling

³

et al. 2019

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“…2.4.8 and 2.4.9] Therefore, a natural question to ask is when can a dessin be defined over its field of moduli. Based on the work of Birch in [2] (see also [32]), a necessary, but not sufficient condition was given in [36] 1 .…”

Section: Invariant 34 (Cartography Group)mentioning

confidence: 99%

“…2.4.8 and 2.4.9] Therefore, a natural question to ask is when can a dessin be defined over its field of moduli. Based on the work of Birch in [2] (see also [32]), a necessary, but not sufficient condition was given in [36] 1 . Theorem 3.2 A dessin can be defined over its field of moduli if there exists a black vertex, or a white vertex, or a face center which is unique for its type and degree.…”

Section: Example 32mentioning

confidence: 99%

Dessins, Their Delta-Matroids and Partial Duals

Malić

¹

2016

Symmetries in Graphs, Maps, and Polytopes

0

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Given a map M on a connected and closed orientable surface, the deltamatroid of M is a combinatorial object associated to M which captures some topological information of the embedding. We explore how delta-matroids associated to dessins behave under the action of the absolute Galois group. Twists of deltamatroids are considered as well; they correspond to the recently introduced operation of partial duality of maps. Furthermore, we prove that every map has a partial dual defined over its field of moduli. A relationship between dessins, partial duals and tropical curves arising from the cartography groups of dessins is observed as well.

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“…Let be as in the introduction. If is singular, or higher-dimensional, it is not true that if , then is defined over its field of moduli (see [SV16, § 5] for singular examples in dimension ). Our main result (Theorem 5.4) is the following: let be a resolution of singularities of .…”

Section: Introductionmentioning

confidence: 99%

Fields of moduli and the arithmetic of tame quotient singularities

Bresciani,

Vistoli

2024

Compositio Math.

3

0

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Given a perfect field $k$ with algebraic closure $\overline {k}$ and a variety $X$ over $\overline {k}$ , the field of moduli of $X$ is the subfield of $\overline {k}$ of elements fixed by field automorphisms $\gamma \in \operatorname {Gal}(\overline {k}/k)$ such that the Galois conjugate $X_{\gamma }$ is isomorphic to $X$ . The field of moduli is contained in all subextensions $k\subset k'\subset \overline {k}$ such that $X$ descends to $k'$ . In this paper, we extend the formalism and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher-dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname {Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli.

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On explicit descent of marked curves and maps

Cited by 7 publications

References 23 publications

A database of Belyi maps

A database of Belyi maps

Dessins, Their Delta-Matroids and Partial Duals

Fields of moduli and the arithmetic of tame quotient singularities

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