Problems in Algebraic Geometry

研究者和他的同事将研究复杂代数几何中的一些问题。首先，他们将探讨一些新的线性级数在射影变量上的渐近不变量，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得到太多关注的基本性质。第二个项目涉及探索从乘数理想中产生的一些新的不变量：这些跳跃数概括了被广泛研究的除数或理想的对数规范阈值，它们似乎编码了相当有趣的几何和代数信息。Lazarsfeld还打算用乘子理想来研究由给定次方程定义的约简射影变的Castelnuovo-Mumford正则性。这个项目的基本焦点（所谓的“代数变量的线性级数”）可以被看作是受几何空间约束的函数族。当空间具有低维时，这种线性序列被用来寻找有效的方法来编码数据，例如在CD上。研究者和他的同事将研究当几何空间具有更高维度时出现的新现象。

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.