Varieties of minimal rational tangents of codimension 1

Jun-Muk Hwang

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 4, page 629-649
  • ISSN: 0012-9593

Abstract

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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 𝒞 x ⊂ 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 𝒞 x is a hypersurface in  ℙ 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.

How to cite

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Hwang, Jun-Muk. "Varieties of minimal rational tangents of codimension 1." Annales scientifiques de l'École Normale Supérieure 46.4 (2013): 629-649. <http://eudml.org/doc/272119>.

@article{Hwang2013,
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.},
author = {Hwang, Jun-Muk},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {varieties of minimal rational tangents; minimal rational curves},
language = {eng},
number = {4},
pages = {629-649},
publisher = {Société mathématique de France},
title = {Varieties of minimal rational tangents of codimension 1},
url = {http://eudml.org/doc/272119},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Hwang, Jun-Muk
TI - Varieties of minimal rational tangents of codimension 1
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 4
SP - 629
EP - 649
AB - 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.
LA - eng
KW - varieties of minimal rational tangents; minimal rational curves
UR - http://eudml.org/doc/272119
ER -

References

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