**Title:**
Linear systems and vanishing theorems

**Time:
June 5-July 2 1998**

**Professor:** Lawrence Ein (University of Illinois at Chicago)

**Collaborator:
**Vlaidimir Masek (Washington University)

**Place:** Department of Mathematics University of Catania

**Scientific program:** Linear systems have played a very important role
in algebraic geometry. In the 50's and 60's, Serre, Kodaira, Grothendieck
Griffiths and others developed the theory of ample line bundles. In the recent
years, linear systems played a central role in the theory of higher dimension
geometry as developed by Mori, Kawamata , Shokurov, Miyaoka, Kollar and others.
Two of the basic tools in this study are the vanishing theorem and the base-points
free theorem. In this course, we will give a survey of these results. We'll
discuss the basic tools. We'll learn about some of the recent progresses and
open questions about effective results concerning linear systems.