Varieties of minimal rational tangents of codimension 1

[Variétés des courbes rationnelles minimales de codimension $1$]

Hwang, Jun-Muk

Annales scientifiques de l'École Normale Supérieure, Tome 46 (2013), p. 629-649 / Harvested from Numdam

Résumé

Soit $X$ une variété projective uniréglée et soit $x$ un point général. D’après le résultat principal de [2], si le degré par rapport à $-K_X$ de toute courbe rationnelle passant par $x$ est au moins égal à $\dim (X)+1$, alors $X$ est un espace projectif. Dans cet article, nous étudions la structure de $X$ sous l’hypothèse que le degré par rapport à $-K_X$ de toute courbe rationnelle passant par $x$ est au moins égal à $\dim (X)$. Notre étude repose sur la variété projective $\mathcal {C}_x\subset \mathbb {P}T_x(X)$ que nous appelons la VMRT (variété des tangentes des courbes rationnelles minimales) en $x$ et qui est définie comme la réunion de toutes les directions tangentes aux courbes rationnelles passant par $x$ dont le degré par rapport à $-K_X$ est minimal. Lorsque ce degré est égal à $\dim (X)$, la VMRT $\mathcal {C}_x$ est une hypersurface de $\mathbb {P}T_x(X)$. Notre résultat principal affirme que si la VMRT en un point général d’une variété projective uniréglée $X$ de dimension $\ge 4$ est une hypersurface, alors $X$ est birationnelle au quotient d’une variété rationnelle explicite par l’action d’un groupe fini. Si, de plus, le rang du groupe de Picard de $X$ est égal à $1$, nous en déduisons que $X$ est une hypersurface quadrique d’un espace projectif.

Abstract

Let $X$ be a uniruled projective manifold and let $x$ be a general point. The main result of [2] says that if the $(-K_X)$-degrees (i.e., the degrees with respect to the anti-canonical bundle of $X$) of all rational curves through $x$ are at least $\dim X +1$, then $X$ is a projective space. In this paper, we study the structure of $X$ when the $(-K_X)$-degrees of all rational curves through $x$ are at least $\dim X$. Our study uses the projective variety ${\mathcal {C}}_x \subset {\mathbb {P}}T_x(X)$, called the VMRT at $x$, defined as the union of tangent directions to the rational curves through $x$ with minimal $(-K_X)$-degree. When the minimal $(-K_X)$-degree of rational curves through $x$ is equal to $\dim X$, the VMRT ${\mathcal {C}}_x$ is a hypersurface in ${\mathbb {P}}T_x(X)$. Our main result says that if the VMRT at a general point of a uniruled projective manifold $X$ of dimension $\ge 4$ is a smooth hypersurface, then $X$ is birational to the quotient of an explicit rational variety by a finite group action. As an application, we show that, if furthermore $X$ has Picard number 1, then $X$ is biregular to a hyperquadric.

Détail

Publié le : 2013-01-01

MR 3098425 | Zbl 1278.14051

DOI : https://doi.org/10.24033/asens.2197
Classification:  14J40,  53B99
Mots clés: variété des tangentes rationnelles minimales, courbes rationnelles minimales

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