November 24, 2007 (Saturday)
Room 517, Meng Wah
Complex, HKU
Professor Stefan
Kebekus
Rational curves on algebraic varieties
Abstract
One approach to
investigate the structure of an algebraic variety X is to study the
geometry of curves, especially the rational curves, that X contains. This
approach relies on classical geometric ideas and strives to understand the
intrinsic geometry of varieties. It is nowadays understood that if X contains
many rational curves, then their geometry determines X to a large degree.
After Shigefumi Mori showed in his landmark works that many
interesting varieties contain rational curves, their systematic study became a
standard tool in algebraic geometry. The spectrum of application is diverse and
covers long-standing problems such as deformation rigidity, stability of the
tangent bundle, classification problems, and generalizations of the Shafarevich
hyperbolicity conjecture.
The expository
lecture concentrates on examples and basic properties of minimal degree
rational curves on projective varieties. Some of the more advanced applications
will be briefly discussed.