arXiv 0809.13862008-09-08math.AGmath.NT

Vari'et'es presque rationnelles, leurs points rationnels et leurs d'eg'en'erescences

J-L. Colliot-Thélène

备注：: Lecture notes for a CIME summer school (Cetraro, September 2007), 56 pages, in French

英文摘要：

This survey, which contains very few proofs, addresses the general question: Over a given type of field, is there a natural class of varieties which automatically have a rational point? Fields under consideration here include: finite fields, p-adic fields, function fields in one or two variables over an algebraically closed field. Classical answers are given by the Chevalley-Warning theorem and by Tsen's theorem. More general answers were provided by a theorem of Graber, Harris and Starr and by a theorem of Esnault. The latter results apply to rationally connected varieties. We discuss these varieties from various angles : weak approximation, R-equivalence, Chow group of zero-cycles. Ongoing work on `rationally simply connected' varieties over function fields in two variables is also mentioned. A common thread in this report is the study of the special fibre of a scheme over a discrete valuation ring: if the generic fibre has a simple geometry, what does it imply for the special fibre?

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

La connexité rationnelle en arithmétique

These expository notes discuss the arithmetic of rationally connected varieties. Detailed proofs of theorems of Kollár, of Kollár and Szabó and of Esnault about rationally connected varieties over finite fields and local fields are given. ----- Nous discutons dans ces notes l'arithmétique des variétés rationnellement connexes. Des preuves détaillées de théorèmes de Kollár, de Kollár et Szabó et d'Esnault concernant les variétés rationnellement connexes sur les corps finis ou locaux y sont données.

2011-01-03math.AGmath.NT

Equivalence rationnelle, equivalence numerique et produits de courbes elliptiques sur un corps fini

This is an update of the first version. We clarify that the main results apply to more general smooth projective varieties X than products of elliptic curves (briefly: X is of "abelian type", e.g. an abelian variety or a product of curves, and the Tate conjecture holds). We also deduce the Gersten conjecture for dvrs whose residue field is the function field of such an X. In an appendix, we construct some functoriality for etale motivic cohomology. Finally, some errors which were in Section 5 of the first version are corrected.

2002-07-20math.AGmath.NT

Produit eulérien motivique et courbes rationnelles sur les variétés toriques

We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin's conjecture about rational points of bounded height on varieties defined over a global field. The study is led through a generating series whose coefficients lie in a Grothendieck ring of motives, the motivic height zeta function. In order to establish convergence properties of this function, we use a notion of eulerian motivic product. It relies on a constructio…

2009-03-26math.AGmath.NT

À propos du groupe fondamental des variétés rationnellement connexes

(On the fundamental group of rationnally connected varieties.) I show that the fundamental group of a normal variety which is rationally chain connected is finite. The proof holds in non-zero characteristic. Je démontre que le groupe fondamental d'une variété normale rationnellement connexe par cha\^ınes est fini. La démonstration est valable en caractéristique différente de zéro.

2003-03-05math.AG

Approximation faible pour les 0-cycles sur un produit de variétés rationnellement connexes

Consider weak approximation for 0-cycles on a smooth proper variety defined over a number field, it is conjectured to be controlled by its Brauer group. Let $X$ be a Châtelet surface or a smooth compactification of a homogeneous space of a connected linear algebraic group with connected stabilizer. Let $Y$ be a rationally connected variety. We prove that weak approximation for 0-cycles on the product $X\times Y$ is controlled by its Brauer group if it is the case for $Y$ after every finite extension of the base field. We do not suppose the existence of 0-cycles of degree $1$ neither on $X$ nor…

2015-03-11math.AGmath.NT

Surfaces stablement rationnelles sur un corps quasi-fini

If a smooth, geometrically rational surface over a finite field is not rational over that field, then over some finite extension of that field the Brauer group of the surface is nonzero. In particular such a surface is not stably rational. This is a special case of a general statement about geometrically rational surfaces which split over a cyclic extension of their field of definition.

2018-06-16math.AG

更多相关论文收起更多论文

非稳定有理的连通实簇

Variétés réelles connexes non stablement rationnelles

本文在实Puiseux级数域上构造了点空间半代数连通却非稳定有理的两个二次曲面光滑交与光滑三次超曲面，而实数域情形仍开放。

2025-06-07math.AG

Automorphismes d'entropie positive, le cas des surfaces rationnelles

A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be a torus, a K3 surface, an Enriques surface or a non-minimal rational surface. We deal with results obtained in this last case. After some recalls on birational maps, algebraic geometry and dynamic, we will speak about Bedford and Kim's works, but also about McMullen's work and more recently about Grivaux and the author's work. These notes have been written for four talks given at IMPA in february/march 2010.

2010-07-07math.AGmath.DS

Courbes rationnelles sur les variétés homogènes et une désingularisation plus fine des variétés de Schubert

In this article we prove the irreducibility of the Hilbert scheme of rationnal curves on homogeneous varieties with fixed class in the Chow ring. This result has also been proved by J. F. Thomsen [T] and B. Kim and R. Pandharipande [KP]. Our method is totaly different (we don't use the compactification of stable maps) and enables us to prove the existence of rational smooth curves on homogeneous varities with fixed class in the Chow ring. This was not the case of Thomsen's and Kim and Pandharipande's proofs. We use a decomposition of G/P in orbits (called the P'-orbits, see definition) which a…

2000-09-15math.AG

The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group)

This is a survey on the ancient question : Let G be a reductive group over an algebraically closed field k and let V be a vector space over k with an almost free linear action of G on V. Let k(V) denote the field of rational functions on V. Is the subfield of G-invariants of k(V) purely transcendental over k ? For G connected, this is still an open question. After a discussion of general matters (various notions of rationality, various notions of quotients, the no-name lemma), we consider several specific groups G. We then discuss the unramified Brauer group of a function field and describe th…

2005-07-07math.AG