Singularities of Pairs and Linear Systems

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This is a project on algebraic geometry. Algebraic geometry studies properties of algebraic varieties, which are geometric objects defined by algebraic equations. Classically algebraic geometers understood the geometry of algebraic curves and algebraic surfaces. But the geometry of algebraic varieties of dimension three or higher remains rather mysterious. One of main difficulties of the problem is that it seems singularities are unavoidable when one studies birational geometry of varieties of dimension three or higher. Ein proposes to use various new technical tools to construct numerical invariants that measure the complexity of these singularities. He also plans to find new applications for these invariants to commutative algebra and birational geometry. In particular, the numerical invariants he studies will play important roles in the study of the Minimal Model Program, which is one the central problems in higher dimensional algebraic geometry.The techniques involved in his investigations include non-classical methods such as the geometry of the arc spaces and multiplier ideals from complex analysis. These numerical invariants also occur naturally in questions on birational rigidity, the theory of D-modules and positive characteristic commutative algebra. While the connections of some these areas are well understood. the appearance of the same invariants in so many different areas is surprising. One of the goals of this proposal is to understand the links of these different aspects better.Algebraic geometry is one of oldest disciplines in mathematics. In recent years, mathematicians have found that there are many important applications of algebraic geometry to mathematical physics, number theory, topology and cryptography. The intellectual impacts of the proposals are on finding new scientific results and gaining a deeper understanding of the geometry of higher dimensional algebraic varieties. In particular, Ein plans to study the singularities that occur naturally in studying higher dimensional birational geometry.