Course Description

Many fundamental properties of a projective algebraic variety X are determined by linear systems |D| of divisors D. Of particular interest are linear systems related to the canonical bundle K_X. For example we would like to understand the linear systems |mK_X| where m is a natural number. One asks whether the linear system gives a birational embedding of X or the so-called Iitaka fibration of X for relatively small numbers m.

 

The aim of these lectures is to give an overview of the subject. We will see that numerous tools and techniques of modern birational geometry naturally show up in the process. For example singularities play a crucial role.