Automorphism groups of spaces of minimal rational curves on Fano manifolds of Picard number
Authors:
JunMuk Hwang and Ngaiming Mok
Translated by:
Journal:
J. Algebraic Geom. 13 (2004), 663673
Published electronically:
February 18, 2004
MathSciNet review:
2072766
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Abstract 
References 
Additional Information
Abstract: Let be a Fano manifold of Picard number and an irreducible component of the space of minimal rational curves on . It is a natural problem to understand the extent to which the geometry of is captured by the geometry of . In this vein we raise the question as to whether the canonical map is an isomorphism. After providing a number of examples showing that this may fail in general, we show that the map is indeed an isomorphism under the additional assumption that the subvariety of consisting of members passing through a general point is irreducible and of dimension .
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Hwang, J.M. and Mok, N.: CartanFubini type extension of holomorphic maps for Fano manifolds of Picard number 1. Journal Math. Pures Appl. 80 (2001) 563575.
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Mukai, S. and Umemura, H.: Minimal rational threefolds. in Algebraic Geometry, Tokyo/Kyoto 1982. Lecture Notes in Math. 1016 (1983) 490518.
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Narasimhan, M.S. and Ramanan, S.: Deformations of the moduli space of vector bundles over an algebraic curve. Ann. of Math. 101 (1975) 391417.
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Semple, J. G. and Roth, L.: Introduction to algebraic geometry. Oxford University Press, 1985.
 [BS]
 Beltrametti, M. and Sommese, A.: The adjunction theory of complex projective varieties. Walter de Gruyter, 1995.
 [DR]
 Desale, U.V. and Ramanan, S.: Classification of vector bundles of rank 2 on hyperelliptic curves. Invent. Math. 38 (1976) 161185.
 [Ho]
 Horikawa, E.: On deformations of holomorphic maps I. J. Math. Soc. Japan 25 (1973) 372396.
 [HM1]
 Hwang, J.M. and Mok, N.: Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation. Invent. Math. 131 (1998) 393418.
 [HM2]
 Hwang, J.M. and Mok, N.: CartanFubini type extension of holomorphic maps for Fano manifolds of Picard number 1. Journal Math. Pures Appl. 80 (2001) 563575.
 [Is]
 Iskovskikh, V.A.: Anticanonical models of 3dimensional algebraic varieties. J. Soviet Math. 13 (1980) 745814.
 [Ke]
 Kebekus, S.: Families of singular rational curves. J. Alg. Geom. 11 (2002) 245256.
 [Ko]
 Kollár, J.: Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, Band 32, SpringerVerlag, 1996.
 [MU]
 Mukai, S. and Umemura, H.: Minimal rational threefolds. in Algebraic Geometry, Tokyo/Kyoto 1982. Lecture Notes in Math. 1016 (1983) 490518.
 [NR]
 Narasimhan, M.S. and Ramanan, S.: Deformations of the moduli space of vector bundles over an algebraic curve. Ann. of Math. 101 (1975) 391417.
 [SR]
 Semple, J. G. and Roth, L.: Introduction to algebraic geometry. Oxford University Press, 1985.
Additional Information
JunMuk Hwang
Affiliation:
Department of Mathematics, Korea Institute for Advanced Study, 20743 Cheongryangridong, Seoul 130012, Korea
Email:
jmhwang@ns.kias.re.kr
Ngaiming Mok
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Email:
nmok@hkucc.hku.hk
DOI:
http://dx.doi.org/10.1090/S1056391104003571
PII:
S 10563911(04)003571
Received by editor(s):
April 9, 2002
Published electronically:
February 18, 2004
Additional Notes:
The first author was supported by Grant No. 980701015L from the KOSEF. The second author was supported by a CERG of the Research Grants Council of Hong Kong
