Introduction: Why Rational Curves?

Abstract

This note is based on a series of lectures given at the Mathematisches Forschungsinstitut at Oberwolfach, Germany, as a part of the DMV seminar “Mori Theory”. The construction of minimal models was discussed by T. Peternell, and my task was to give an overview of various aspects of the study of rational curves on algebraic varieties, including the following topics:

• (a) Techniques which enable us to find rational curves on certain classes of varieties;

• (b) Characterization of uniruled varieties (varieties that carry sufficiently many rational curves) in terms of canonical divisors;

• (c) Generic semipositivity of the cotangent bundle of non-uniruled varieties and its application to the “abundance conjecture” in dimension three;

• (d) Decomposition of a given variety into the “non-uniruled part” and the “rationally connected part”,

• (e) Application of the techniques above to the theory of Fano varieties.