Abstract
The aim of these notes is to give an introduction to the ideas and techniques of handling rational curves on varieties. The main emphasis is on varieties with many rational curves, which are the higher dimensional analogs of rational curves and surfaces.
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References
V. Alexeev, Moduli spaces M g,n (W) for surfaces, Higher-dimensional complex varieties (Trento. 1994), de Gruyter (Berlin, 1996), pp. 1–22.
M. Artin and G. Winters, Degenerate fibres and stable reduction of curves, Topology, 10 (1971), 373–383.
C. H. Clemens and P. Griffiths, The Intermediate Jacobian of the cubic three-fold, Ann. of Math., 95 (1972), 281–356.
P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. INES, 36 (1969). 75–109.
T. Graber. J. Harris and J. Starr. Families of Rationally Connected Varieties. J. Arner. Math. Soc, 16 (2003), 29–55.
A. Grothendieck. Technique de construction et théoremes d’existence en géométrie algébrique IV: Les schémas de Hilbert, Séminaire Bourbaki, 1960–1961, exposé 221, Soc. Math. France (Paris, 1962).
A. Grothendieck, Technique de descente et théoremes d’existence en géométrie algébrique V. Les schémas de Picard: théoremes dexistence; VI. Les schémas de Picard: propriétés générates, Séminaire Bourbaki, Vol. 7 exposés 232, 236, Soc. Math. France (Paris, 1962).
W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in: Algebraic geometry—Santa Cruz 1995 (S. Bloch et al., eds.) Proc. Sympos. Pure Math., vol. 62, Part 2, Arner. Math. Soc. (Providence, RI, 1997), pp. 45–96.
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer (1977).
S. Keel and S. Mori. Quotients by groupoids, Ann. of Math., 145 (1997), 193–213.
J. Kollár, Projectivity of complete moduli. J. Differential Geom., 32 (1990), 235–268.
J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 32, Springer (1996).
J. Kollár, Quotient spaces modulo algebraic groups, Ann. of Math., 145 (1997), 33–79.
J. Kollár, Rationally connected varieties over local fields, Ann. of Math., 150 (1999), 357–367.
J. Kollár, Which are the simplest Algebraic Varieties?, Bull. Amer. Math. Soc., 38 (2001), 409–433.
J. Kollár, Specialization of zero cycles, preprint math. AG/0205148.
J. Kollár, Y. Miyaoka and S. Mori, Rationally Connected Varieties, J. Algebraic Geom., 1 (1992), 429–448.
J. Kollár and S. Mori, Birational geometry of algebraic varieties, with the collaboration of C. II. Clemens and A. Corti. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press (1998).
M. Kontsevich and Y. I. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Coram. Math. Phys., 164 (1994), 525–562.
A. F. Lopez, Noether-Lefschetz Theory and the Picard Group of Projective Surfaces, Mem. Arner. Math. Soc., vol. 89 (1991).
G. Xu, Subvarieties of General Hypersurfaces in Projective Space, J. Differential Geom., 39 (1994), 139–172.
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Araujo, C., Kollár, J. (2003). Rational Curves on Varieties. In: Böröczky, K., Kollár, J., Szamuely, T. (eds) Higher Dimensional Varieties and Rational Points. Bolyai Society Mathematical Studies, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05123-8_3
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DOI: https://doi.org/10.1007/978-3-662-05123-8_3
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