Felix Klein Lectures

The Equations Defining Projective Varieties

Videos can be found on the him lectures youtube channel.

Speaker: Robert Lazarsfeld (Stony Brook University)

Date: January 6-22, 2014; Tuesdays and Thursdays, 10:15-12:00

Venue: HIM lecture hall, Poppelsdorfer Allee 45

Abstract:

Over the past thirty years there has been considerable interest in questions related in one way or another to the algebraic properties of projective varieties. On the one hand, classical results about defining equations of curves and abelian varieties have emerged as the first cases of more general statements for higher syzygies, and the picture for general smooth varieties has started to come into focus. In terms of a natural measure of algebraic complexity, a fascinating but still mysterious dichotomy has emerged between the behavior of nonsingular varieties and arbitrary schemes.

Although the questions are algebraic in origin, a wide variety of sophisticated geometric tools have been used to study them. These include for example the geometry of vector bundles, vanishing theorems for multiplier ideals, Fourier-Mukai transforms on abelian varieties and the geometry of Hilbert schemes.

This lecture series will attempt to provide a broad survey of this circle of ideas, aimed particularly at geometrically-oriented algebraic geometers. I plan to organize the lectures around the algebraic questions, but to focus particularly on the various geometric tools that have been used to study them. I will also discuss many open problems in the area.