Problems in Algebraic Geometry

代数几何问题

• 批准号: 0139713
• 负责人: Robert Lazarsfeld
• 金额: $ 34.5万
• 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: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0139713&HistoricalAwards=false
• 关键词: Problems; Algebraic; Geometry

The Investigator and his colleagues will study a number of problems in complex algebraic geometry. First, they will explore some new asymptotic invariants of linear series on a projective variety, especially their variational behavior. While the geometry of linear series is a very classical topic, new methods have the potential to shed light on basic properties that have not up to now received much attention. A second project involves the exploration of some new invariants arising from multiplier ideals: these jumping numbers generalize the much-studied log-canonical threshold of a divisor or ideal, and they seem to encode quite interesting geometric and algebraic information. Lazarsfeld also intends to use multiplier ideals to study the Castelnuovo-Mumford regularity of reduced projective varieties defined by equations of given degree.The basic focus of this project (so-called "linear series on algebraic varieties") can be viewed as families of functions constrained by geometric spaces. When the spaces have low dimension such linear series have been used to find efficient ways to encode data, for example on CD's. The Investigator and his colleagues will study the new phenomena that arise when the geometric spaces have higher dimension.

研究人员及其同事将研究复杂代数几何形状的许多问题。首先，他们将探索一些有关射影变化的线性系列的新渐近不变，尤其是它们的变化行为。虽然线性系列的几何形状是一个非常古典的主题，但新方法有可能阐明尚未引起很多关注的基本属性。第二个项目涉及对乘数理想引起的一些新不变的探索：这些跳跃数字概括了分数或理想的备受研究的对数 - 典型的阈值，它们似乎编码了相当有趣的几何和代数信息。 Lazarsfeld还打算使用乘数理想来研究由给定程度方程定义的降低的投射品种的Castelnuovo-Mumford规律性。该项目的基本重点（所谓的“代数品种上的线性系列”）可以看作是由几何空间限制的功能家族。当空间具有较低的尺寸时，该线性系列已被用来寻找有效的编码数据的方法，例如在CD上。研究人员及其同事将研究几何空间具有较高维度时出现的新现象。