The Arithmetic of Tame Quotient Singularities in Dimension <i>2</i>

Bresciani, Giulio

doi:10.1093/imrn/rnad079

Cited by 4 publications

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References 9 publications

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“…Note that subgroups and quotients of groups are not necessarily : for example, is , but is not. Furthermore, the product of two groups is not necessarily , see [Bre24, Remark 18] for a counterexample with .…”

Section: The Arithmetic Of Tame Quotient Singularitiesmentioning

confidence: 99%

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Fields of moduli and the arithmetic of tame quotient singularities

Bresciani,

Vistoli

2024

Compositio Math.

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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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Section: The Arithmetic Of Tame Quotient Singularitiesmentioning

confidence: 99%

“…There is much more that one could say about groups. The paper [Bre24] by the first author contains a complete classification of groups. The case seems much harder; hopefully it will be the subject of further work.…”

Section: Introductionmentioning

confidence: 99%

Fields of moduli and the arithmetic of tame quotient singularities

Bresciani,

Vistoli

2024

Compositio Math.

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The field of moduli of sets of points in $$\mathbb {P}^{2}$$

Bresciani

2024

Arch. Math.

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For every $$n\ge 6$$ n ≥ 6 , we give an example of a finite subset of $$\mathbb {P}^{2}$$ P 2 of degree n which does not descend to any Brauer–Severi surface over the field of moduli. Conversely, for every $$n\le 5$$ n ≤ 5 , we prove that a finite subset of degree n always descends to a 0-cycle on $$\mathbb {P}^{2}$$ P 2 over the field of moduli.

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An arithmetic valuative criterion for proper maps of tame algebraic stacks

Bresciani

Vistoli

2023

manuscripta math.

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The valuative criterion for proper maps of schemes has many applications in arithmetic, e.g. specializing $$\mathbb {Q}_{p}$$ Q p -points to $$\mathbb {F}_{p}$$ F p -points. For algebraic stacks, the usual valuative criterion for proper maps is ill-suited for these kind of arguments, since it only gives a specialization point defined over an extension of the residue field, e.g. a $$\mathbb {Q}_{p}$$ Q p -point will specialize to an $$\mathbb {F}_{p^{n}}$$ F p n -point for some n. We give a new valuative criterion for proper maps of tame stacks which solves this problem and is well-suited for arithmetic applications. As a consequence, we prove that the Lang–Nishimura theorem holds for tame stacks.

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The Arithmetic of Tame Quotient Singularities in Dimension 2

Cited by 4 publications

References 9 publications

Fields of moduli and the arithmetic of tame quotient singularities

Fields of moduli and the arithmetic of tame quotient singularities

The field of moduli of sets of points in $$\mathbb {P}^{2}$$

An arithmetic valuative criterion for proper maps of tame algebraic stacks

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