arXiv subjects
Publications and source records attributed to Jean-Louis Colliot-Thélène.
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.
In a smooth family of projective, complex varieties, stable rationality need not be preserved under generisation. This was proved by Hassett, Pirutka and Tschinkel upon use of the specialisation method. Work of Schreieder produced many more examples and introduced a simplification of the specialisation method (no explicit desingularisation). In this text, I try to describe the combined method from scratch in one of the simplest cases. A small, further simplification consists in using specialisation of R-equivalence in place of Fulton's specialisation for the Chow group of zero-cycles.
This is a survey on (lack of) stable rationality over arbitrary fields (including algebraically closed fields). Topics addressed include: Rationality and unirationality, R-equivalence on rational points, Chow groups of zero-cycles, Galois action on the Picard group, Brauer group, higher unramified cohomology, global differentials, specialisation method (via R-equivalence), geometrically rational surfaces, cubic hypersurfaces.
Let $k$ be a field and $X \subset P^3_{k}$ a smooth cubic surface. Let $Δ\subset Pic(X)$ be the finite index subgroup spanned by norms of lines on $X_{K}$ for $K$ running through the finite separable extensions of $k$. The quotient $Pic(X)/Δ$ is 3-primary. If $X$ contains a line defined over $k$, then $Δ=Pic(X)$.
We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. In particular, we show that local-global principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the function field. This assertion is analogous to a known result concerning varieties over number fields. We also show that our results hold more generally in the henselian case.
We establish the vanishing of the third unramified cohomology group for many types of Fano hypersurfaces in projective space over an algebraically closed field of arbitrary characteristic, and over a finite field. For cubic hypersurfaces over a finite field, the case of fourfolds remains open. --- Sur un corps algébriquement clos et sur un corps fini, on établit de nouveaux résultats d'annulation pour la cohomologie non ramifiée de degré 3 pour de nombreux types d'hypersurfaces de Fano. Le cas des hypersurfaces cubiques de dimension 4 sur un corps fini reste ouvert.
Over any complex cubic hypersurface of dimension at least 2, the Chow group of 1-dimensional cycles is spanned by the lines lying on the hypersurface. The smooth case has already been given several other proofs. -- On montre que sur toute hypersurface cubique complexe de dimension au moins 2, le groupe de Chow des cycles de dimension 1 est engendré par les droites. Le cas lisse est un théorème connu. La démonstration ici donnée repose sur un résultat sur les surfaces géométriquement rationnelles sur un corps quelconque (1983), obtenu via la K-théorie algébrique.
Toute hypersurface cubique lisse complexe de dimension au moins 2 dont l'équation est donnée par l'annulation d'une somme de formes cubiques à variables séparées, chaque forme impliquant au plus trois variables, est universellement $CH_{0}$-triviale. --- If a smooth cubic hypersurface of dimension at least 2 is defined by the vanishing of a sum of cubic forms in independent variables and each of these forms involves at most 3 variables, then the cubic hypersurface is universally $CH_{0}$-trivial : there is an integral Chow decomposition of the diagonal.
We propose and compare various techiques available to produce smooth cubic hypersurfaces over a non-algebraically-closed field which have rational points but which are not stably rational over their ground field.
Let K be the function field of a curve over the complex field. Let X be a homogeneous space of a semisimple linear algebraic group. Strong approximation holds for X outside any finite nonempty set of places of K. Strong approximation fails for tori over K.
En appliquant des méthodes développées par Kollár, Voisin, nous-mêmes, Totaro, nous montrons qu'un revêtement cyclique de $\mathbb P_{\mathbb C}^n, n\geq 3$ de degré premier $p$, ramifié le long d'une hypersurface très générale de degré $mp$ n'est pas stablement rationnel si $m(p-1) <n+1\leq mp$. En basse dimension, on retrouve le cas des revêtements doubles de $\mathbb P^3_{\mathbb C},$ ramifiés le long d'une quartique (Voisin) et des revêtements doubles de $\mathbb P^3_{\mathbb C}$ ramifiés le long d'une sextique (Beauville), et l'on obtient aussi les revêtements doubles de $\mathbb P^4_{\mathbb C}$ ramifiés le long d'une sextique. La méthode produit des exemples définis sur un corps de nombres. Using the methods developed by Kollár, Voisin, ourselves, Totaro, we prove that a cyclic cover of $\mathbb P^n_{\mathbb C}, n\geq 3$ of prime degree $p$ ramified along a very general hypersurface of degree $mp$ is not stably rational if $m(p-1) <n+1\leq mp$. In small dimensions, we recover double covers of $\mathbb P^3_{\mathbb C}$, ramified along a quartic (Voisin), and double covers of $\mathbb P^3_{\mathbb C}$, ramified along a sextic (Beauville), and we also find double covers of $\mathbb P^4_{\mathbb C}$, ramified along a sextic. This method also allows one to produce examples over a number field.
Let C be the complex field and K=C((x,y)) or K=C((x))(y). Let G be a connected linear algebraic group over K. Under the assumption that the K-variety G is K-rational, i.e. that the function field is purely transcendant, it was proved that a principal homogeneous space of G has a rational point over K as soon as it has one over each completion of K with respect to a discrete valuation. In this paper we show that one cannot in general do without the K-rationality assumption. To produce our examples, we introduce a new type of obstruction. It is based on higher reciprocity laws on a 2-dimensional scheme. We also produce a family of principal homogeneous spaces for which the refined obstruction controls exactly the existence of rational points.
We construct a conic bundle over an elliptic curve over a real quadratic field that is a counterexample to the Hasse principle not explained by the étale Brauer-Manin obstruction. We also give simple examples of threefolds with the same property that are families of 2-dimensional quadrics, and discuss some other examples and general properties of the Brauer-Manin obstruction.
Sur toute surface de del Pezzo de degré 4 sur un corps $C_1$ de caractéristique zéro, tous les points rationnels sont R-équivalents. Plus généralement, ceci vaut sur tout corps parfait infini de caractéristique différente de 2. ---- On a del Pezzo surface of degree 4 over a characteristic zero $C_1$-field, all rational points are R-equivalent.This more generally holds over any infinite perfect $C_{1}$-field of characteristic different from 2.
Connections between the second Chow group of a smooth projective variety and its third unramified cohomology group, with coefficients the roots of unity twisted twice, feature in several recent works. In this note we revisit a 1996 paper by B. Kahn and specialize it to various types of rationally connected varieties.
Inspirés par un argument de C. Voisin, nous montrons l'existence d'hypersurfaces quartiques lisses dans ${\bf P}^4_{\mathbb C}$ qui ne sont pas stablement rationnelles, plus précisément dont le groupe de Chow de degré zéro n'est pas universellement égal à $\mathbb Z$. --- There are (many) smooth quartic hypersurfaces in ${\bf P}^4_{\mathbb C}$ which are not stably rational. More precisely, their degree zero Chow group is not universally equal to $\mathbb Z$. The proof uses a variation of a specialisation method due to C. Voisin.
In arXiv:1410.5671, Jahnel and Loughran prove the local global principle for existence of linear spaces of dimension $r$ on smooth complete intersections of two quadrics in projective space of dimension $2r+2$. We present an alternative proof of a slightly more general result. ----- Dans arXiv:1410.5671, Jahnel et Loughran établissent un principe local-global pour l'existence d'espaces linéaires de dimension $r$ dans les intersections complètes lisses de deux quadriques dans un espace projectif de dimension $2r+2$. On donne une démonstration alternative d'un résultat un peu plus général.
Pour $K$ un corps global (corps de nombres ou corps de fonctions d'une variable sur un corps fini $F$), on dispose de théorèmes de dualité classiques (Tate, Poitou, Nakayama) pour la cohomologie galoisienne à valeurs dans des tores et des modules abéliens finis. Nous établissons de tels théorèmes pour $K$ le corps des fonctions d'une courbe projective et lisse sur le corps $F={\bf C}((t))$ des séries formelles en une variable sur le corps des complexes. Cela permet de contrôler le défaut du principe de Hasse et l'approximation faible pour les espaces homogènes sous un tore. Il y a ici des différences avec le cas classique ($K$ corps global), et aussi avec le cas récemment étudié où $K$ est un corps de fonctions d'une variable sur un corps $p$-adique. Par exemple, la $K$-rationalité d'un tore n'implique pas ici la validité du principe de Hasse pour ses espaces principaux homogènes. ---------------- Over a global field $K$ (number field, or function field of a curve over a finite field $F$), arithmetic duality theorems for the Galois cohomology of tori and finite Galois modules have long been known. More recent work investigates the case where $K$ is the function fields of of a curve over a $p$-adic field. For $K$ the function field of a curve over the formal series field $F={\bf C}((t))$, we establish analogous duality theorems. We thus control the obstruction to the local-global principle and to weak approximation for homogeneous spaces of tori. There are differences with the afore described cases. For example the Hasse principle need not hold for principal homogeneous spaces of a $K$-rational torus.