Varieties of minimal rational tangents of codimension 1
Annales scientifiques de l'École Normale Supérieure, Série 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 (-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 dimX+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 dimX. Our study uses the projective variety 𝒞 x 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 dimX, the VMRT 𝒞 x is a hypersurface in T x (X). Our main result says that if the VMRT at a general point of a uniruled projective manifold X of dimension 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(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 𝒞 x 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 𝒞 x est une hypersurface de T x (X). Notre résultat principal affirme que si la VMRT en un point général d’une variété projective uniréglée X de dimension 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, Série 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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