Rational Connectedness and Galois Covers of the Projective Line

Colliot-Thélène, Jean-Louis

doi:10.2307/121121

Cited by 31 publications

(18 citation statements)

References 12 publications

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“…That is, if an irreducible L-variety has a smooth L-point, then the L-points are Zariski dense. This property of p-closed fields was noticed by J.-L. Colliot-Thélène [CT00] in the case where L is perfect. In the case where L is not perfect it was proved by M. Jarden [Jar03].…”

Section: P-versalitysupporting

confidence: 56%

Versality of algebraic group actions and rational points on twisted varieties

Duncan

Reichstein

2015

J. Algebraic Geom.

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We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic variety X. Our main result is that these notions of versality are equivalent to various statements concerning rational points on twisted forms of X (existence of rational points, existence of a dense set of rational points, etc.). We give applications of this equivalence in both directions to study versality of group actions and rational points on algebraic varieties. We obtain similar results on p-versality for a prime integer p. An appendix, containing a letter from J.-P. Serre, puts the notion of versality in a historical perspective. 2.4. It is vital that algebraic groups are linear throughout this paper. For example, if A = {1} is an abelian variety, then no versal A-torsor can exist. Otherwise the exponent of every element of H 1 (K, A), for any field K/k, would divide the exponent of the versal torsor. However, there are A-torsors of arbitrarily high exponent (over suitable K); cf. [Bro07, Section 3].Proposition 2.5. If X is a versal irreducible G-variety, then X is geometrically irreducible.Proof. It suffices to show that X k s is irreducible, where k s denotes a separable closure of k; see [Har77, Exercise 2.3.15(a)]. Let X 1 , . . . , X n denote the irreducible components of X k s . We want to show that n = 1. We will argue by contradiction. Assume n 2. Since X is irreducible over k, the absolute Galois group Gal(k) permutes X 1 , . . . , X n transitively. Thus Y := X 1 ∩ · · · ∩ X n is a closed G-invariant k-subvariety of X (possibly empty) and X = Y .Let V be a generically free linear G-representation with k(V ) G infinite. By Remark 2.1 there exists a G-equivariant rational k-map f : V X \ Y . Since V is geometrically irreducible, the image of f is contained in one of the components X i . Since Gal(k) transitively permutes the components, the image of f is also contained in X 2 , . . . , X n and thus in Y , a contradiction.Remark 2.6. Suppose X is of minimal dimension among generically free versal G-varieties. Then X must be very versal. To see this, let V be a generically free linear representation of G, as in Remark 2.3. Let U ⊂ X be a G-invariant open subvariety which is the total space of a G-torsor U → B. Then U is weakly versal and, by Remark 2.1, there exists a G-equivariant rational map f : V U . The closure Z of the image of f is very versal and, since U is a G-torsor, the action on Z is generically free. By minimality of dim(X), we conclude that dim(Z) = dim(X) and thus f is dominant.The minimal value of dim(X)−dim(G), as X ranges over the very versal (or equivalently, versal) generically free G-varieties, is called the essential dimension of G and is denoted by ed(G); see [BF03], [Ser03, Section 5] or [Rei10].Remark 2.7. The following criterion gives a convenient way (often the only known way) to show that a given action is not versal.Let X be a projective irreducible weakly versal G-variety defined over k. Suppose H ⊂ G is a closed k-subgroup such that every finite-dimen...

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Section: P-versalitysupporting

confidence: 56%

Versality of algebraic group actions and rational points on twisted varieties

Duncan

Reichstein

2015

J. Algebraic Geom.

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“…cit., and proved to be the "right class" of fields over which one can do a lot of interesting mathematics, like (inverse) Galois theory, see e.g. Colliot-Thélène [CT00], Pop [Pop96], and the survey article Harbater [Har03], study torsors of finite groups Moret-Bailly [MB01], study rationally connected varieties Kollár [Kol99], study the elementary theory of function fields , etc. 1 Recall that a field K is called a large field, if Supported by NSF grants DMS-0401056 and DMS-0801144.…”

Section: Introductionmentioning

confidence: 99%

Henselian implies large

Pop

2010

Ann. Math.

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“…Recall that a field k is called p-closed, if every finite extension of k has degree a power of p. In particular, the absolute Galois group G k of k is a pro-p group, and further: If p = char(k), then k is perfect, whereas if p = char(k), then k might also have purely inseparable extensions. To the best of my knowledge, Colliot-Thélèn [7] was the first to notice that p-closed fields are large, see also Jarden [34], Ch. 5, and Pfister, [51] for more on p-closed fields.…”

Section: C) the P-closed Fieldsmentioning

confidence: 99%

“…The notion of large field was introduced in Pop [59] and proved to be the "right class" of fields over which one can do a lot of interesting mathematics, like (inverse) Galois theory, see Colliot-Thélène [7], Moret-Bailly [45], Pop [59], [61], the survey article Harbater [30], study torsors of finite groups Moret-Bailly [46], study rationally connected varieties Kollár [37], study the elementary theory of function fields Koenigsmann [35], Poonen-Pop [55], characterize extremal valued fields as introduced by Ershov [12], see Azgin-Kuhlmann-Pop [1], etc. Maybe that is why the "large fields" acquired several other names -google it:épais, fertile, weite Körper, ample, anti-Mordellic.…”

Section: Introductionmentioning

confidence: 99%

Little survey on large fields - old & new

Pop

2014

Valuation Theory in Interaction

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The large fields were introduced by the author in [59] and subsequently acquired several other names. This little survey includes earlier and new developments, and at the end of each section we mention a few open questions.We fix notations to be used throughout the paper and recall a few basic facts as follows: Let k be an arbitrary field. A k-variety is any separated reduced k-scheme of finite type, and a k-curve is a k-variety of dimension one. For every k-variety X and every field extension l|k we denote by X(l) the l-rational points of X, hence X(k) is the set of k-rational points of X. Given a k-variety X, we denote by X sm and X reg the smooth, respectively regular, locus of X. We recall that X sm and X reg are open k-subvarieties

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Rational Connectedness and Galois Covers of the Projective Line

Cited by 31 publications

References 12 publications

Versality of algebraic group actions and rational points on twisted varieties

Versality of algebraic group actions and rational points on twisted varieties

Henselian implies large

Little survey on large fields - old & new

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