Problems in Algebraic Geometry

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Lazarsfeld will work on a number of problems in complex algebraic geometry. First, with Mustata, he will investigate in detail a procedure by which one associates a convex body to a linear series on a projective variety. Introduced in passing by Okounkov, this construction has the potential to shed important new light on the structure of linear systems. A second set of problems involves using multiplier ideals and related tools from higher dimensional geometry to study some questions of an essentially local nature. Specifically, Lazarsfeld hopes to resolve some conjectures of De Fernex, Ein and Mustata relating algebraic invariants of an ideal, and to prove in all dimensions a result about attenuation of singularities of graded families of ideals suggested by a theorem of Favre-Jonsson concerning currents in the plane. Algebraic geometry, one of the oldest and most central fields of mathematics, deals with the geometric study of the solutions of systems of polynomial equations. It touches on many other fields of mathematics, ranging from number theory to topology, algebra and complex analysis. It has found important applications to problems in such diverse areas as coding theory and theoretical physics. The particular questions that Lazarsfeld will study involve relating questions in algebraic geometry to geometric properties of solid bodies in space and to special collections of polynomials. It is hoped that this work will lead to the development of some valuable new techniques in the field.