Let $X$ be a rationally connected algebraic variety, defined over a number field $k.$ We find a relation between the arithmetic of rational points on $X$ and the arithmetic of zero-cycles. More precisely, we consider the following statements: (1) the Brauer-Manin obstruction is the only obstruction to weak approximation for $K$-rational points on $X_K$ for all finite extensions $K/k;$ (2) the Brauer-Manin obstruction is the only obstruction to weak approximation in some sense that we define for zero-cycles of degree...

The product of two Schubert classes in the quantum $K$-theory ring of a homogeneous space $X=G/P$ is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on $X$. We show that if $X$ is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to $X$ that take the marked points to general Schubert varieties and whose domains...

A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered...

It has been previously established that a Cremona transformation of bidegree (2,2) is linearly equivalent to the projectivization of the inverse map of a rank 3 Jordan algebra. We call this result the “$JC$-correspondence”. In this article, we apply it to the study of quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension 4 and 5.

La méthode de la descente a été introduite et développée par Colliot-Thélène et Sansuc. Elle permet d’étudier l’arithmétique de certaines variétés rationnelles. Dans ce texte on montre comment il en résulte que pour certaines familles $f:X\to Y$ de variétés rationnelles sur un corps local $k$ de caractéristique nulle le nombre des classes de $R$-équivalence de la fibre $X_y\left(k\right)$ est localement constant quand $y$ varie dans $Y\left(k\right)$.

Soit $r\ge 1$, $n\ge 2$, et $q\ge n-1$ des entiers. On introduit la classe ${𝒳}_{r+1,n}\left(q\right)$ des sous-variétés $X$ de dimension $r+1$ d’un espace projectif, telles que pour $(x_1,...,x_n)\in X^n$ générique, il existe une courbe rationnelle normale de degré $q$, contenue dans $X$ et passant par les points $x_1,...,x_n$ ; $X$ engendre un espace projectif dont la dimension, pour $r$, $n$ et $q$ donnés, est la plus grande possible compte tenu de la première propriété. Sous l’hypothèse $q\ne 2n-3$, on détermine toutes les variétés $X$ appartenant à la classe ${𝒳}_{r+1,n}\left(q\right)$. On montre en particulier qu’il existe une...

Let $X$ be a normal projective variety, and let $A$ be an ample Cartier divisor on $X$. Suppose that $X$ is not the projective space. We prove that the twisted cotangent sheaf ${\Omega }_X\otimes A$ is generically nef with respect to the polarisation $A$. As an application we prove a Kobayashi-Ochiai theorem for foliations: if $ℱ⊊T_X$ is a foliation such that $detℱ\equiv i_{ℱ}A$, then $i_{ℱ}$ is at most the rank of $ℱ$.