Higher Dimensional Algebraic Geometry

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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 surfaces. But the geometry of varieties of dimension three or higher remains rather mysterious. The goals of this project are to study the properties of the equations defining these varieties and investigate the properties of the spaces of cycles. Another main difficulty of studying higher dimensional varieties 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 study the invariants that measure the complexity of these singularities. The new techniques involve the geometry of the arc spaces, generic limits 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. The appearance of the same invariants in so many different areas of mathematics is surprising. One of the goals of this proposal is to understand the links of these different aspects better.Algebraic geometry is one of the 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 proposal 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 syzygies, spaces of higher co-dimensional cycles and the singularities that occur naturally in studying higher dimensional birational geometry.