Problems in Algebraic Geometry

拉扎斯菲尔德将研究复杂代数几何中的一些问题。首先，与Mustata一起，他将详细研究将凸体与射影变化上的线性级数联系起来的过程。顺便介绍一下Okounkov，这种结构有可能为线性系统的结构提供重要的新线索。第二组问题涉及使用乘数理想和高维几何中的相关工具来研究一些本质上是局部性质的问题。具体来说，Lazarsfeld希望解决De Fernex、Ein和Mustata关于理想的代数不变量的一些猜想，并在所有维度上证明Favre-Jonsson关于平面电流的定理所提出的关于梯度理想族奇点衰减的结果。代数几何是数学中最古老和最核心的领域之一，涉及多项式方程系统解的几何研究。它涉及数学的许多其他领域，从数论到拓扑学，代数和复分析。它在编码理论和理论物理等不同领域的问题中得到了重要的应用。Lazarsfeld将研究的特殊问题涉及到代数几何中的问题与空间中固体的几何性质以及多项式的特殊集合的关系。希望这项工作将导致该领域一些有价值的新技术的发展。

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.