# Varieties of minimal rational tangents of codimension 1

Annales scientifiques de l'École Normale Supérieure (2013)

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

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topHwang, 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] C. Araujo, Rational curves of minimal degree and characterizations of projective spaces, Math. Ann.335 (2006), 937–951. Zbl1109.14032MR2232023
- [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] 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] 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] 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] 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] 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] J.-M. Hwang & N. Mok, Birationality of the tangent map for minimal rational curves, Asian J. Math.8 (2004), 51–63. Zbl1072.14015MR2128297
- [9] S. Kebekus, Families of singular rational curves, J. Algebraic Geom.11 (2002), 245–256. Zbl1054.14035MR1874114
- [10] S. Kobayashi & T. Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ.13 (1973), 31–47. Zbl0261.32013MR316745
- [11] J. Kollár, Rational curves on algebraic varieties, Ergebn. Math. Grenzg. 32, Springer, 1996. Zbl0877.14012
- [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] 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] C. Okonek, M. Schneider & H. Spindler, Vector bundles on complex projective spaces, Progress in Math. 3, Birkhäuser, 1980. Zbl0438.32016MR561910
- [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

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