Pseudo-reflection groups and essential dimension

Duncan, Alexander; Reichstein, Zinovy

doi:10.1112/jlms/jdu056

Cited by 8 publications

(4 citation statements)

References 37 publications

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“…• Alexander Duncan and Zinovy Reichstein observed that Theorem 1.6 can be used to extend Theorem 1.4 of their article [DR14] to the case of an arbitrary ground field k; originally they proved it only over an infinite k. Their theorem compares variants of the notion of essential dimension for finite subgroups of GL n (k). Here is another application, in the same spirit as [DR14, Theorem 8.1].…”

Section: Applicationsmentioning

confidence: 99%

Bertini irreducibility theorems over finite fields

Charles

Poonen

2014

J. Amer. Math. Soc.

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Given a geometrically irreducible subscheme X in P^n over F_q of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that the intersection of H and X is geometrically irreducible tends to 1 as d tends to infinity. We also prove variants in which X is over an extension of F_q, and in which the immersion of X in P^n is replaced by a more general morphism.Comment: 14 pages. In the previous version, in the proof of Lemma 5.1 we forgot to reduce to the case of a normal variety before implicitly using what is called Lemma 3.6 in this version. We fixed this by inserting Lemma 3.6 and adjusting the proof of Lemma 5.1 (and the hypotheses in Lemma 5.2

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Section: Applicationsmentioning

confidence: 99%

Bertini irreducibility theorems over finite fields

Charles

Poonen

2014

J. Amer. Math. Soc.

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“…Let W (E 6 ) act on h via the defining representation, and let A(h) denote the corresponding faithful linear W (E 6 )-variety. Then by [DuRe2,Lemma 6.1]…”

Section: Finding a Single Linementioning

confidence: 99%

Resolvent degree, Hilbert’s 13th Problem and geometry

Farb

Wolfson

2020

Enseign. Math.

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We develop the theory of resolvent degree, introduced by Brauer [Bra2] in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a finite group. As one application of this point of view, we prove that Hilbert's 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.

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“…To see this, we need the following Bertini-type result which asserts the existence of general hypersurface sections containing a given closed subvariety. See [22,Theorem 1] or [10,Theorem 8.1] for a similar treatment. For the convenience of the reader, here we give its proof.…”

Section: Weak Numerical Equivalencesmentioning

confidence: 99%

A theorem of Tits type for automorphism groups of projective varieties in arbitrary characteristic

2019

Math. Ann.

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We show an analogue of Jordan's theorem for algebraic groups defined over a field of arbitrary characteristic. As a consequence, a Jordan-type property holds for the automorphism group of any projective variety over . However, the above theorem is false for fields of characteristic p > 0 due to the existence of unipotent elements of finite order. For instance, the group GL n (F p ) contains arbitrarily large subgroups of the form SL n (F p r ) which are simple modulo their centers. Nevertheless, for any finite subgroup Γ of GL n ( ) of order not divisible by char( ), there still exists a normal abelian subgroup A of Γ with [Γ : A] ≤ J(n) for the same J(n) as in Theorem 1.

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Pseudo-reflection groups and essential dimension

Cited by 8 publications

References 37 publications

Bertini irreducibility theorems over finite fields

Bertini irreducibility theorems over finite fields

Resolvent degree, Hilbert’s 13th Problem and geometry

A theorem of Tits type for automorphism groups of projective varieties in arbitrary characteristic

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