Problems in Algebraic Geometry

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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.