An arithmetic valuative criterion for proper maps of tame algebraic stacks

Bresciani, Giulio; Vistoli, Angelo

doi:10.1007/s00229-023-01491-6

Cited by 6 publications

(3 citation statements)

References 10 publications

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“…In this proof, and in the rest of the paper, a crucial role is played by the Lang-Nishimura theorem for tame stacks that we prove in [BV23]. For the convenience of the reader we recall its statement.…”

Section: The Main Resultsmentioning

confidence: 93%

“…Theorem 5.5 [BV23,Theorem 4.1]. Let S be a scheme and X Y a rational map of algebraic stacks over S, with X locally noetherian and integral and Y tame and proper over S. Let k be a field, s : Spec k → S a morphism.…”

Section: The Main Resultsmentioning

confidence: 99%

“…For the second and third part, the fundamental tool is the Lang-Nishimura theorem for tame stacks proved in [BV23] (see Theorem 5.5).…”

Section: Fields Of Moduli and The Arithmetic Of Tame Quotient Singula...mentioning

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 Main Resultsmentioning

confidence: 93%

Section: The Main Resultsmentioning

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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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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On the birational section conjecture with strong birationality assumptions

Bresciani

2023

Invent. math.

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Let $X$ X be a curve over a field $k$ k finitely generated over ℚ and $t$ t an indeterminate. We prove that, if $s$ s is a section of $\pi _{1}(X)\to \operatorname{Gal}(k)$ π 1 ( X ) → Gal ( k ) such that the base change $s_{k(t)}$ s k ( t ) is birationally liftable, then $s$ s comes from geometry. As a consequence we prove that the section conjecture is equivalent to the cuspidalization of all sections over all finitely generated fields.

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An arithmetic valuative criterion for proper maps of tame algebraic stacks

Cited by 6 publications

References 10 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 $$\mathbb {P}^{2}$$

On the birational section conjecture with strong birationality assumptions

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