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 defined by algebraic equations. Algebraic geometry has important connections to mathematical physics and coding theory. The research will improve understanding of the relations between the geometry of algebraic curves and higher dimensional varieties and the set of higher relations among the equations defining the varieties. These relations are called the syzygies of the varieties. The investigator and collaborators will investigate the properties of the syzygies of algebraic curves and higher dimensional varieties. They will study singularities of higher dimensional varieties and their relations to the minimal model program and other branches of mathematics. The project has extensive training components as well. The PI has already trained several successful young algebraic geometers and is currently supervising two graduate students who are working on various problems related to the project. The PI and co-authors are writing a book on multiplier ideals, extension theorems, and geometry of arc spaces. The book aims to help young researchers understand how these techniques can be applied to understanding higher dimensional algebraic geometry.The research project involves study of syzygies by studying properties of the Hilbert schemes of points on the curve. It is also planned to investigate the asymptotic behavior of syzygies of an algebraic variety and their relations to the intrinsic geometry of the variety. The project would extend the classical results previously only known in the case of algebraic curves to higher dimensional varieties. Multiplier ideals together with their vanishing theorems play an important role in the development of the higher dimensional minimal model program; these ideals are also an important measure of the complexity of singularities. Mather-Jacobian ideals have an advantage over the classical multiplier ideals because these ideals are defined for non-Q-Gorenstein singularities. Higher co-dimension effective cycles should be very useful in understanding the geometry of higher dimensional varieties. The investigator and collaborators are planning to study numerical invariants such as log-canonical thresholds and minimal log-discrepancies attached to the singularities of algebraic varieties. These invariants play central roles in the study of minimal model program. They also show up naturally in the theory of D-modules, positive characteristic commutative algebra, and the geometry of arc spaces.