Problems in Algebraic Geometry

代数几何问题

• 批准号: 0652845
• 负责人: Robert Lazarsfeld
• 金额: $ 47.37万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652845&HistoricalAwards=false
• 关键词: Problems; Algebraic; Geometry

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.

拉扎斯菲尔德将致力于解决复杂代数几何中的许多问题。首先，他将与 Mustata 一起详细研究将凸体与射影簇上的线性级数关联起来的过程。这种结构由奥孔科夫顺便提出，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的相关工具来研究一些本质上局部性质的问题。具体来说，拉扎斯菲尔德希望解决德·费内克斯、艾因和穆斯塔塔关于理想代数不变量的一些猜想，并在所有维度上证明关于平面电流的法夫尔-琼森定理所提出的理想分级族奇点衰减的结果。代数几何是最古老和最核心的数学领域之一，涉及多项式方程组解的几何研究。它涉及数学的许多其他领域，从数论到拓扑、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上都有重要的应用。拉扎斯菲尔德将研究的特定问题涉及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。