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References
Carolina Araujo and János Kollár. Rational curves on varieties. In Higher dimensional varieties and rational points (Budapest, 2001), volume 12 of Bolyai Soc. Math. Stud., pages 13–68. Springer, Berlin, 2003.
Sergei J. Arakelov. Families of algebraic curves with fixed degeneracies. Izv. Akad. Nauk SSSR Ser. Mat., 35:1269–1293, 1971.
Carolina Araujo. Rationally connected varieties. In Snowbird lectures in algebraic geometry, volume 388 of Contemp. Math., pages 1–16. Amer. Math. Soc., Providence, RI, 2005.
Sebastien Boucksom, Jean-Pierre Demailly, Mihai Păun, and Thomas Peternell. The pseudo-effectuve cone of a compact Kähler manifold and varieties of negative Kodaira dimension. preprint math.AG/0405285, May 2004.
Arnaud Beauville. Riemanian holonomy and algebraic geometry. preprint math.AG/9902110, 1999.
Fedor A. Bogomolov and Michael L. McQuillan. Rational curves on foliated varieties. IHES Preprint, 2001.
Armand Borel and Raghavan Narasimhan. Uniqueness conditions for certain holomorphic mappings. Invent. Math., 2:247–255, 1967.
Jean-Benoît Bost. Algebraic leaves of algebraic foliations over number fields. Publ. Math. Inst. Hautes Études Sci., (93):161–221, 2001.
Frédéric Campana. Connexité rationnelle des variétés de Fano. Ann. Sci. École Norm. Sup. (4), 25(5):539–545, 1992.
Luca Chiantini and Ciro Ciliberto. Weakly defective varieties. Trans. Amer. Math. Soc., 354(1):151–178 (electronic), 2002.
Herbert Clemens, János Kollár, and Shigefumi Mori. Higher-dimensional complex geometry. Astérisque, (166):144 pp. (1989), 1988.
César Camacho and Alcides Lins Neto. Geometric theory of foliations. Birkhäuser Boston Inc., Boston, MA, 1985. Translated from the Portuguese by Sue E. Goodman.
Koji Cho, Yoichi Miyaoka, and Nicholas I. Shepherd-Barron. Characterizations of projective space and applications to complex symplectic manifolds. In Higher dimensional birational geometry (Kyoto, 1997), volume 35 of Adv. Stud. Pure Math., pages 1–88. Math. Soc. Japan, Tokyo, 2002.
Koji Cho and Ei-ichi Sato. Smooth projective varieties with the ample vector bundle Λ2 T X in any characteristic. J. Math. Kyoto Univ., 35(1):1–33, 1995.
Olivier Debarre. Higher-dimensional algebraic geometry. Universitext. Springer-Verlag, New York, 2001.
Hubert Flenner. Restrictions of semistable bundles on projective varieties. Comment. Math. Helv., 59(4):635–650, 1984.
Tom Graber, Joe Harris, and Jason Starr. Families of rationally connected varieties. J. Amer. Math. Soc., 16(1):57–67 (electronic), 2003.
Alexander Grothendieck. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math., (28):255, 1966.
Rajendra V. Gurjar and De-Qi Zhang. π1 of smooth points of a log del Pezzo surface is finite. I. J. Math. Sci. Univ. Tokyo, 1(1):137–180, 1994.
Robin Hartshorne. Cohomological dimension of algebraic varieties. Ann. of Math. (2), 88:403–450, 1968.
Robin Hartshorne. Ample vector bundles on curves. Nagoya Math. J., 43:73–89, 1971.
Joe Harris. Algebraic geometry, volume 133 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. A first course, Corrected reprint of the 1992 original.
Jun-Muk Hwang and Stefan Kebekus. Geometry of chains of minimal rational curves. J. Reine Angew. Math., 584:173–194, 2005.
Jun-Muk Hwang, Stefan Kebekus, and Thomas Peternell. Holomorphic maps onto varieties of non-negative Kodaira dimension. J. Alg. Geom., posted April 21, 2005, PII S 1056-3911(05)00411-X (to appear in print). Preprint math.AG/0307220, Juli 2003.
Daniel Huybrechts and Manfred Lehn. The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig, 1997.
Jun-Muk Hwang and Ngaiming Mok. Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation. Invent. Math., 131(2):393–418, 1998.
Jun-Muk Hwang and Ngaiming Mok. Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds. Invent. Math., 136(1):209–231, 1999.
Jun-Muk Hwang and Ngaiming Mok. Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1. J. Math. Pures Appl. (9), 80(6):563–575, 2001.
Jun-Muk Hwang and Ngaiming Mok. Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles. J. Algebraic Geom., 12(4):627–651, 2003.
Jun-Muk Hwang and Ngaiming Mok. Birationality of the tangent map for minimal rational curves. Asian J. Math., 8(1):51–63, 2004.
Andreas Höring. Uniruled varieties with splitting tangent bundle. preprint math.AG/0505327, May 2005.
Jun-Muk Hwang. Rigidity of homogeneous contact manifolds under Fano deformation. J. Reine Angew. Math., 486:153–163, 1997.
Jun-Muk Hwang. Stability of tangent bundles of low-dimensional Fano manifolds with Picard number 1. Math. Ann., 312(4):599–606, 1998.
Jun-Muk Hwang. Tangent vectors to Hecke curves on the moduli space of rank 2 bundles over an algebraic curve. Duke Math. J., 101(1):179–187, 2000.
Jun-Muk Hwang. Geometry of minimal rational curves on Fano manifolds. In School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), volume 6 of ICTP Lect. Notes, pages 335–393. Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001. Available on the ICTP web site at http://www.ictp.trieste.it/~pub_off/services.
Jun-Muk Hwang. Hecke curves on the moduli space of vector bundles over an algebraic curve. In Algebraic geometry in East Asia (Kyoto, 2001), pages 155–164. World Sci. Publishing, River Edge, NJ, 2002.
Shigeru Iitaka. Algebraic geometry, volume 76 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1982. An introduction to birational geometry of algebraic varieties, North-Holland Mathematical Library, 24.
Stefan Kebekus. Lines on contact manifolds. J. Reine Angew. Math., 539:167–177, 2001.
Stefan Kebekus. Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron. In Complex geometry (Göttingen, 2000), pages 147–155. Springer, Berlin, 2002.
Stefan Kebekus. Families of singular rational curves. J. Algebraic Geom., 11(2):245–256, 2002.
Stefan Kebekus. Projective bundles of singular plane cubics. Math. Nachr., 242:119–131, 2002.
Stefan Kebekus. Holomorphe Abbildungen auf Mannigfaltigkeiten mit nicht-negativer Kodaira-Dimension. In Y. Tschinkel, editor, Mathematisches Institut Georg-August-Universität Göttingen Seminars 2003/2004, pages 157–166. Universitätsverlag der Georg-August-Universität Göttingen, 2004.
Stefan Kebekus. Lines on complex contact manifolds. II. Compos. Math., 141(1):227–252, 2005.
Stefan Kebekus and Sándor J. Kovács. Are rational curves determined by tangent vectors? Ann. Inst. Fourier (Grenoble), 54(1):53–79, 2004.
Stefan Kebekus and Sándor J. Kovács. Families of canonically polarized a a varieties over surfaces. preprint math.AG/0511378, November 2005. To appear in Invent. Math.
János Kollár and Shigefumi Mori. Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.
Seán Keel and James McKernan. Rational curves on quasi-projective surfaces. Mem. Amer. Math. Soc., 140(669):viii+153, 1999.
János Kollár, Yoichi Miyaoka, and Shigefumi Mori. Rational connectedness and boundedness of Fano manifolds. J. Differential Geom., 36(3):765–779, 1992.
János Kollár, Yoichi Miyaoka, and Shigefumi Mori. Rationally connected varieties. J. Algebraic Geom., 1(3):429–448, 1992.
János Kollár. Extremal rays on smooth threefolds. Ann. Sci. École Norm. Sup. (4), 24(3):339–361, 1991.
János Kollár, editor. Flips and abundance for algebraic threefolds. Société Mathématique de France, Paris, 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992).
János Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin, 1996.
Sándor J. Kovács. Algebraic hyperbolicity of fine moduli spaces. J. Algebraic Geom., 9(1):165–174, 2000.
Sándor J. Kovács. Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties. Compositio Math., 131(3):291–317, 2002.
Sándor J. Kovács. Families of varieties of general type: the Shafarevich conjecture and related problems. In Higher dimensional varieties and rational points (Budapest, 2001), volume 12 of Bolyai Soc. Math. Stud., pages 133–167. Springer, Berlin, 2003.
Stefan Kebekus and Thomas Peternell. A refinement of Stein factorization and deformations of surjective morphisms. preprint math.AG/0508285, August 2005.
Stefan Kebekus, Thomas Peternell, Andrew J. Sommese, and Jarosław A. Wiśniewski. Projective contact manifolds. Invent. Math., 142(1):1–15, 2000.
Stefan Kebekus, Luis Solá Conde, and Matei Toma. Rationally connected a foliations after Bogomolov and McQuillan. J. Alg. Geom., posted on May 25, 2006, PII S 1056-3911(06)00435-8 (to appear in print).
Adrian Langer. Addendum to: “Semistable sheaves in positive characteristic” [Ann. of Math. (2) 159 (2004), no. 1, 251–276; mr 2051393]. Ann. of Math. (2), 160(3):1211–1213, 2004.
Adrian Langer. Semistable sheaves in positive characteristic. Ann. of Math. (2), 159(1):251–276, 2004.
Robert Lazarsfeld. A Barth-type theorem for branched coverings of projective space. Math. Ann., 249(2):153–162, 1980.
Robert Lazarsfeld. Positivity in algebraic geometry. II, volume 49 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals.
Kenji Matsuki. Introduction to the Mori program. Universitext. Springer-Verlag, New York, 2002.
Yoichi Miyaoka. Deformations of a morphism along a foliation and applications. In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), volume 46 of Proc. Sympos. Pure Math., pages 245–268. Amer. Math. Soc., Providence, RI, 1987.
Yoichi Miyaoka and Shigefumi Mori. A numerical criterion for uniruledness. Ann. of Math. (2), 124(1):65–69, 1986.
Shigefumi Mori. Projective manifolds with ample tangent bundles. Ann. of Math. (2), 110(3):593–606, 1979.
Shigefumi Mori. Threefolds whose canonical bundles are not numerically effective. Ann. of Math. (2), 116(1):133–176, 1982.
Yoichi Miyaoka and Thomas Peternell. Geometry of higher-dimensional algebraic varieties, volume 26 of DMV Seminar. Birkhäuser Verlag, Basel, 1997.
Vikram B. Mehta and Annamalai Ramanathan. Semistable sheaves on projective varieties and their restriction to curves. Math. Ann., 258(3):213–224, 1981/82.
Masayoshi Miyanishi and Shuichiro Tsunoda. The structure of open algebraic surfaces. II. In Classification of algebraic and analytic manifolds (Katata, 1982), volume 39 of Progr. Math., pages 499–544. Birkhäuser Boston, Boston, MA, 1983.
Masayoshi Miyanishi and Shuichiro Tsunoda. Noncomplete algebraic surfaces with logarithmic Kodaira dimension ∔ ∞ and with nonconnected boundaries at infinity. Japan. J. Math. (N.S.), 10(2):195–242, 1984.
Alan M. Nadel. The boundedness of degree of Fano varieties with Picard number one. J. Amer. Math. Soc., 4(4):681–692, 1991.
Mudumbai S. Narasimhan and S. Ramanan. Geometry of Hecke cycles. I. In C. P. Ramanujam—a tribute, volume 8 of Tata Inst. Fund. Res. Studies in Math., pages 291–345. Springer, Berlin, 1978.
A. N. Parshin. Algebraic curves over function fields. I. Izv. Akad. Nauk SSSR Ser. Mat., 32:1191–1219, 1968.
Thomas Peternell and Andrew J. Sommese. Ample vector bundles and branched coverings. Comm. Algebra, 28(12):5573–5599, 2000. With an appendix by Robert Lazarsfeld, Special issue in honor of Robin Hartshorne.
Miles Reid. Young person’ guide to canonical singularities. In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), volume 46 of Proc. Sympos. Pure Math., pages 345–414. Amer. Math. Soc., Providence, RI, 1987.
C. S. Seshadri. Fibrés vectoriels sur les courbes algébriques, volume 96 of Astérisque. Société Mathématique de France, Paris, 1982. Notes written by J.-M. Drezet from a course at the école Normale Supérieure, June 1980.
Igor R. Shafarevich. Algebraic number fields. In Proc. Internat. Congr. Mathematicians (Stockholm, 1962), pages 163–176. Inst. Mittag-Leffler, Djursholm, 1963. English translation: Amer. Math. Soc. Transl. (2) 31 (1963), 25–39.
Yum-Tong Siu. Hyperbolicity in complex geometry. In The legacy of Niels Henrik Abel, pages 543–566. Springer, Berlin, 2004.
Xiaotao Sun. Minimal rational curves on moduli spaces of stable bundles. Math. Ann., 331(4):925–937, 2005.
Eckart Viehweg. Positivity of direct image sheaves and applications to families of higher dimensional manifolds. In School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), volume 6 of ICTP Lect. Notes, pages 249–284. Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001. Available on the ICTP web site at http://www.ictp.trieste.it/~pub_off/services.
Eckart Viehweg and Kang Zuo. On the isotriviality of families of projective manifolds over curves. J. Algebraic Geom., 10(4):781–799, 2001.
Eckart Viehweg and Kang Zuo. Base spaces of non-isotrivial families of smooth minimal models. In Complex geometry (Göttingen, 2000), pages 279–328. Springer, Berlin, 2002.
Fyodor L. Zak. Tangents and secants of algebraic varieties, volume 127 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1993. Translated from the Russian manuscript by the author.
De-Qi Zhang. Logarithmic del Pezzo surfaces of rank one with contractible boundaries. Osaka J. Math., 25(2):461–497, 1988.
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Kebekus, S., Conde, L.S. (2006). Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications. In: Catanese, F., Esnault, H., Huckleberry, A.T., Hulek, K., Peternell, T. (eds) Global Aspects of Complex Geometry. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-35480-8_10
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