A Survey of Extremal Rays for Fano Manifolds

Main Article Content

Toru Tsukioka*

Abstract

In this expository note, we give an introduction to algebraic geometry with the emphasis on Fanomanifolds which forms one of the building blocks of algebraic varieties. After a brief review on the theory of intersection numbers, we introduce numerical invariants on Fano manifolds and discuss related problems.


Keywords: algebraicvarieties, Fano manifolds, extremal ray


E-mail: [email protected]

Article Details

Section
Original Research Articles

References

[1] Beauville, A. 1983.Complex algebraic surfaces. London Mathematical Society Student Texts34, Cambridge University Press
[2] Casagrande, C. 2012. On the Picard number of divisors in Fano manifolds. The Annales Scientifiques de l’Ecole Normale Superieure.
[3] Kollar, J. 1996. Rational curves on algebraic varieties. Ergebnisseders Mathematik und ihrerGrenzgebiete 3. Folge. Band 32, Springer
[4] Kollar, J. and Mori, S. 1998. Birational geometry of algebraic varieties. Cambridge tracts in Mathematics 134, Cambridge University Press
[5] Lazarsfeld, R. 2004. Positivity in algebraic geometry I. ErgebnissederMathematik und ihrerGrenzgebiete 3. Folge. Band 48, Springer
[6] Hartshorne, R. 1977. Algebraic geometry. Graduate Texts in Mathematics 52, Springer
[7] Iitaka, S. 1972. Genus and classification of algebraic varieties. I. Sugaku 24, 14-27
[8] Parchin, A. and Shavarevich, I. 1999. Algebraic geometry V. Encyclopaedia of Mathematical Sciences 47, Springer
[9] Shavarevich, I. 1988. Basic Algebraic Geometry 1. Springer.
[10] Hwang, J.-M. 1998. Some Recent Topics in Fano Manifolds. Trends in Mathematics Information Center for Mathematical Sciences 1, 26–30
[11] Tsukioka, T. 2012. On the minimal length of extremal rays for Fano four-folds. Math. Zeit. 271, 555–564