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.

调查员和他的同事们将研究一些复杂的代数几何问题。首先，他们将探索射影簇上线性级数的一些新的渐近不变量，特别是它们的变分行为。虽然线性级数的几何是一个非常经典的话题，但新的方法有可能揭示迄今为止尚未受到太多关注的基本性质。第二个项目涉及探索一些新的不变量所产生的乘数理想：这些跳跃的数字推广了大量研究的对数规范阈值的除数或理想，他们似乎编码相当有趣的几何和代数信息。Lazarsfeld还打算使用乘子理想研究Castelnuovo-Mumford正则性的减少投影品种所定义的方程给degree.The基本重点的项目（所谓的“线性系列代数品种”）可以被视为家庭的职能限制的几何空间。当空间具有低维度时，这种线性序列已经被用于找到对数据进行编码的有效方式，例如在CD上。研究者和他的同事们将研究几何空间具有更高维度时出现的新现象。