Rational Curves on Algebraic Varieties

The aim of this article is to give a brief review on recent developments in the theory of embedded rational curves, which the author believes is a new, useful viewpoint in the study of higher dimensional algebraic varieties. By an embedded rational curve, or simply a rational curve, on a variety X, we mean the image of the projective line ℙ1 by a nontrivial morphism to X, hence complete, one-dimensional, but not necessarily smooth.

[1]  E. Brieskorn.  Ein Satz über die komplexen Quadriken , 1964 .

[2]  C. B. Allendoerfer,et al.  Hyperbolic manifolds and holomorphic mappings , 1970 .

[3]  Shôshichi Kobayashi,et al.  Characterizations of complex projective spaces and hyperquadrics , 1973 .

[4]  R. Brody,et al.  A family of smooth hyperbolic hypersurfaces in $\mathbb{P}_3$ , 1977 .

[5]  V. A. Iskovskih.  Fano 3-FOLDS. I , 1977 .

[6]  Shigefumi Mori,et al.  Proj ective manifolds with ample tangent bundles , 1979 .

[7]  F. Bogomolov.  Holomorphic tensors and vector bundles on projective varieties , 1979 .

[8]  S. Yau,et al.  Compact kähler manifolds of positive bisectional curvature , 1980 .

[9]  Y. Siu.  Curvature characterization of hyperquadrics , 1980 .

[10]  M. Levine.  Deformations of uni-ruled varieties , 1981 .

[11]  S. Mukai,et al.  Classification of Fano 3-folds with B2≥2 , 1981 .

[12]  A. Fujiki.  Deformation of Uniruled Manifolds , 1981 .

[13]  J. Kollár,et al.  Riemann-Roch Type Inequalities , 1983 .

[14]  A. Beauville,et al.  Variétés Kähleriennes dont la première classe de Chern est nulle , 1983 .

[15]  Miles Reid,et al.  Young person''s guide to canonical singularities , 1985 .

[16]  Y. Miyaoka,et al.  A numerical criterion for uniruledness , 1986 .

[17]  K. Narain.  New Heterotic String Theories in Uncompactified Dimensions < 10 , 1986 .

[18]  Y. Miyaoka.  The Chern Classes and Kodaira Dimension of a Minimal Variety , 1987 .

[19]  Shigefumi Mori,et al.  Flip theorem and the existence of minimal models for 3-folds , 1988 .

[20]  J. Kollár,et al.  Higher dimensional complex geome-try , 1988 .

[21]  Y. Miyaoka.  Abundance conjecture for 3-folds : case $\nu = 1$ , 1988 .

[22]  Yoichi Miyaoka,et al.  On the Kodaira dimension of minimal threefolds , 1988 .

[23]  E. Viehweg.  Weak positivity and the stability of certain Hilbert points , 1990 .

[24]  Lorenzo Ramero.  Effective estimates for unirationality , 1990 .

[25]  A. Nadel.  The boundedness of degree of Fano varieties with Picard number one , 1991 .

[26]  F. Campana,et al.  Connexité rationnelle des variétés de Fano , 1992 .

[27]  J. Kollár.  Flips and Abundance for Algebraic Threefolds , 1992 .

[28]  K. Cho,et al.  Projective manifold with the ample vector bundle $^2_\Lambda \mathrm{T_X}$ , 1992 .

[29]  J. Kollár,et al.  Rational connectedness and bound-edness of Fano manifolds , 1992 .

[30]  Yujiro Kawamata,et al.  Abundance theorem for minimal threefolds , 1992 .

[31]  J. Kollár,et al.  Rational curves on Fano varieties , 1992 .

[32]  F. Campana,et al.  Projective threefolds containing a smooth rational surface with ample normal bundle. , 1993 .

[33]  Formal neighbourhoods of a toric variety and unirationality of algebraic varieties , 1994 .

[34]  Y. Ruan,et al.  A mathematical theory of quantum cohomology , 1994 .

[35]  János Kollár,et al.  Rational curves on algebraic varieties , 1995, Ergebnisse der Mathematik und ihrer Grenzgebiete.

[36]  János Kollár,et al.  Rationally Connected Varieties , 1996 .