Varieties of minimal rational tangents of codimension 1
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 4, p. 629-649

Let $X$ be a uniruled projective manifold and let $x$ be a general point. The main result of [2] says that if the $\left(-{K}_{X}\right)$-degrees (i.e., the degrees with respect to the anti-canonical bundle of $X$) of all rational curves through $x$ are at least $dimX+1$, then $X$ is a projective space. In this paper, we study the structure of $X$ when the $\left(-{K}_{X}\right)$-degrees of all rational curves through $x$ are at least $dimX$. Our study uses the projective variety ${𝒞}_{x}\subset ℙ{T}_{x}\left(X\right)$, called the VMRT at $x$, defined as the union of tangent directions to the rational curves through $x$ with minimal $\left(-{K}_{X}\right)$-degree. When the minimal $\left(-{K}_{X}\right)$-degree of rational curves through $x$ is equal to $dimX$, the VMRT ${𝒞}_{x}$ is a hypersurface in $ℙ{T}_{x}\left(X\right)$. 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.

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 $dim\left(X\right)+1$, 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 $dim\left(X\right)$. Notre étude repose sur la variété projective ${𝒞}_{x}\subset ℙ{T}_{x}\left(X\right)$ 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 $dim\left(X\right)$, la VMRT ${𝒞}_{x}$ est une hypersurface de $ℙ{T}_{x}\left(X\right)$. 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.

DOI : https://doi.org/10.24033/asens.2197
Classification:  14J40,  53B99
Keywords: varieties of minimal rational tangents, minimal rational curves
@article{ASENS_2013_4_46_4_629_0,
author = {Hwang, Jun-Muk},
title = {Varieties of minimal rational tangents of codimension 1},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {Ser. 4, 46},
number = {4},
year = {2013},
pages = {629-649},
doi = {10.24033/asens.2197},
zbl = {1278.14051},
mrnumber = {3098425},
language = {en},
url = {http://www.numdam.org/item/ASENS_2013_4_46_4_629_0}
}

Hwang, Jun-Muk. Varieties of minimal rational tangents of codimension 1. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 4, pp. 629-649. doi : 10.24033/asens.2197. http://www.numdam.org/item/ASENS_2013_4_46_4_629_0/

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