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

top
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 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  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

top

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

top
  1. [1] C. Araujo, Rational curves of minimal degree and characterizations of projective spaces, Math. Ann.335 (2006), 937–951. Zbl1109.14032MR2232023
  2. [2] K. Cho, Y. Miyaoka & N. I. Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds, in Higher dimensional birational geometry (Kyoto, 1997), Adv. Stud. Pure Math. 35, Math. Soc. Japan, 2002, 1–88. Zbl1063.14065MR1929792
  3. [3] J. Hong & J.-M. Hwang, Characterization of the rational homogeneous space associated to a long simple root by its variety of minimal rational tangents, in Algebraic geometry in East Asia—Hanoi 2005, Adv. Stud. Pure Math. 50, Math. Soc. Japan, 2008, 217–236. Zbl1186.14044MR2409558
  4. [4] J.-M. Hwang, Geometry of minimal rational curves on Fano manifolds, in School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect. Notes 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, 335–393. MR1919462
  5. [5] J.-M. Hwang, Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents, Ann. Sci. Éc. Norm. Supér. 43 (2010), 607–620. Zbl1210.14044MR2722510
  6. [6] J.-M. Hwang & N. Mok, Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds, Invent. Math.136 (1999), 209–231. Zbl0963.32007MR1681093
  7. [7] J.-M. Hwang & N. Mok, Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1, J. Math. Pures Appl.80 (2001), 563–575. Zbl1033.32013MR1842290
  8. [8] J.-M. Hwang & N. Mok, Birationality of the tangent map for minimal rational curves, Asian J. Math.8 (2004), 51–63. Zbl1072.14015MR2128297
  9. [9] S. Kebekus, Families of singular rational curves, J. Algebraic Geom.11 (2002), 245–256. Zbl1054.14035MR1874114
  10. [10] S. Kobayashi & T. Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ.13 (1973), 31–47. Zbl0261.32013MR316745
  11. [11] J. Kollár, Rational curves on algebraic varieties, Ergebn. Math. Grenzg. 32, Springer, 1996. Zbl0877.14012
  12. [12] Y. Miyaoka, Numerical characterisations of hyperquadrics, in Complex analysis in several variables—Memorial Conference of Kiyoshi Oka’s Centennial Birthday, Adv. Stud. Pure Math. 42, Math. Soc. Japan, 2004, 209–235. MR2087053
  13. [13] N. Mok, Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents, in Third International Congress of Chinese Mathematicians. Parts 1, 2, AMS/IP Stud. Adv. Math. 42, Amer. Math. Soc., 2008, 41–61. Zbl1182.14042MR2409622
  14. [14] C. Okonek, M. Schneider & H. Spindler, Vector bundles on complex projective spaces, Progress in Math. 3, Birkhäuser, 1980. Zbl0438.32016MR561910
  15. [15] K. Yamaguchi, Differential systems associated with simple graded Lie algebras, in Progress in differential geometry, Adv. Stud. Pure Math. 22, Math. Soc. Japan, 1993, 413–494. Zbl0812.17018MR1274961

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.