Topics in Algebraic Geometry

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This is a project on algebraic geometry, which is one of the oldest disciplines in mathematics. It studies geometric objects called varieties that are defined by algebraic equations. Algebraic geometry has important connections to mathematical physics and coding theory. The intellectual impacts of the proposal are on improving our understanding on the relations between the geometry of algebraic curves and higher dimensional varieties to the set of relations among the equations defining the varieties. These relations are called the syzygies of the varieties. The PI will investigate the properties of the syzygies of algebraic curves and higher dimensional varieties. The PI also plans to study the irrationality of an algebraic variety, a measure of the complexity of the variety. This project has extensive educational and training components.The PI and his collaborator Lazarsfeld will study syzygies by of the curves using the geometric properties of the subvarieties of the Hilbert schemes of points on the curve. They will investigate the asymptotic behavior of syzygies of an algebraic varieties and their relations to the intrinsic geometry of the variety. The proposed program would extend the classical results previously only known in the case of algebraic curves to higher dimensional varieties. Another part of this project is the study of the irrationality of an algebraic variety with emphasis on the case when the variety is a very general polarized K3 surface. Together with Xudong Zheng (Johns Hopkins University), the PI will study the Hilbert schemes of points for singular surfaces; in particular, the relations between the singularities of the surfaces and the geometry of Hilbert schemes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.