The behavior of essential dimension under specialization II

Reichstein, Zinovy; Scavia, Federico

doi:10.48550/arxiv.2112.13298

Cited by 2 publications

(3 citation statements)

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“…Theorem 1.2 also implies a slightly weaker version of Theorem 1.1, in which the morphism X → Spec k is assumed to be smooth. Theorems 1.2 and 1.3 are related in spirit with the results in [16,17].…”

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confidence: 77%

The genericity theorem for the essential dimension of tame stacks

Bresciani

Vistoli

2022

Pure and Applied Mathematics Quarterly

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Let X be a regular tame stack. If X is locally of finite type over a field, we prove that the essential dimension of X is equal to its generic essential dimension; this generalizes a previous result of P. Brosnan, Z. Reichstein and the second author. Now suppose that X is locally of finite type over a 1-dimensional noetherian local domain R with fraction field K and residue field k. We prove that ed Introduction, and the statement of the main theoremsLet k be a field, X → Spec k an algebraic stack, an extension of k, ξ ∈ X( ) an object of X over . If k ⊆ L ⊆ is an intermediate extension, we say, very naturally, that L is a field of definition of ξ if ξ descends to L. The essential dimension ed k ξ, which is either a natural number or +∞, is the minimal transcendence degree of a field of definition of ξ. If X is of finite type then ed k ξ is always finite.This number ed k ξ is a very natural invariant, which measures, essentially, the number of independent parameters that are needed for defining ξ. The essential dimension ed k X of X is the supremum of the essential dimension of all objects over all extensions of k (if X is empty then ed k X is −∞). This number is the answer to the question "how many independent parameters

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confidence: 77%

The genericity theorem for the essential dimension of tame stacks

Bresciani

Vistoli

2022

Pure and Applied Mathematics Quarterly

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“…The horizontal map is injective by [29,Lemma 3.3(b)]. 4 Note that the assumptions of Theorem 1.3, that G 0 is reductive and G/G 0 is finite over D (and hence, over R and over S), are used to ensure that [29, Lemma 3.3(b)] applies.…”

Section: The Resolvent Degree Of An Abelian Varietymentioning

confidence: 99%

“…[29, Lemma 3.3(b)] is a variant of the Grothendieck-Serre Conjecture over a Henselian discrete valuation ring. A theorem of Nisnevich, establishing the Grothendick-Serre Conjecture in this context, is a key ingredient in our proof of both Proposition 13.1 and Theorem 1.3.…”

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confidence: 99%

Hilbert's 13th Problem for Algebraic Groups

Reichstein¹

2022

Preprint

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The algebraic form of Hilbert's 13th Problem asks for the resolvent degree rd(n) of the general polynomial f (x) = x n +a 1 x n−1 +. . .+a n of degree n, where a 1 , . . . , a n are independent variables. The resolvent degree is the minimal integer d such that every root of f (x) can be obtained in a finite number of steps, starting with C(a 1 , . . . , a n ) and adjoining algebraic functions in d variables at each step. Recently Farb and Wolfson defined the resolvent degree rd k (G) of any finite group G and any base field k of characteristic 0. In this setting rd(n) = rd C (S n ), where S n denotes the symmetric group. In this paper we define rd k (G) for every algebraic group G over an arbitrary field k, investigate the dependency of this quantity on k and show that rd k (G) 5 for any field k and any connected group G. The question of whether rd k (G) can be bigger than 1 for any field k and any algebraic group G over k (not necessarily connected) remains open.

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The behavior of essential dimension under specialization II

Cited by 2 publications

References 7 publications

The genericity theorem for the essential dimension of tame stacks

The genericity theorem for the essential dimension of tame stacks

Hilbert's 13th Problem for Algebraic Groups

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