Linear Systems on Higher Dimensional Varieties

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

The principal investigator proposes to study linear systems of higher dimensional varieties. In particular, he plans to apply the methods of higher dimensional geometry to study some basic questions from different areas of algebraic geometry such as birational geometry, commutative algebra, computational complexity, geometry of algebraic curves, jet schemes of singular varieties and geometry of big divisors. He has already obtained results in the following areas, (1) effective results on Nullstellensatz and syzygies of ideals, (2) singularities of theta divisors and birational geometry of irregular varieties, (3) Fujita's problems on adjoint linear systems and (4) studying symbolic power of an ideal. But there are still many fundamental problems remain unsolved. He plans to develop new methods and techniques to study them. Algebraic geometry is one of the oldest areas in mathematics and it studies geometric, algebraic and arithmetic properties of geometric objects defined by algebraic equations. Algebraic geoemtry also has applications to mathematical physics and cryptography. In this project, the principal investigator develops new techniques to study algebra, computational complexity and properties of higher dimensional geometric objects.