Singularities and Duality with Applications to Moduli Theory

University Of Washington, Seattle WA

Investigators

Abstract

This project is in the field of algebraic geometry, one of the oldest fields in modern mathematics, and one that blossomed to the point where it has solved centuries-old problems. In its simplest form it treats figures defined in space by polynomials, such as a sphere or an (infinite) cone. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover, it has proved applicable in fields as diverse as physics, theoretical computer science, cryptography, coding theory, and robotics. A central problem in algebraic geometry is the classification of all geometric objects. In turn, an important part of classification theory is the theory of moduli. The latter's core idea is that one not only wants to understand these objects, but also understand the way they can be deformed. Moduli spaces play an important role in theoretical physics: studying curves on moduli spaces provides information on how an object is moving in space-time. The investigator is also involved in mentoring the next generation of researchers. This project supports several graduate students, who are working toward their doctoral degree on related projects under the direction of the investigator. The project concerns several topics in higher dimensional algebraic geometry, especially singularities and their applications to moduli theory. At the center is the study of singularities and their interconnections, especially rational, Du Bois, and other singularities related to the minimal model program and moduli theory of higher dimensional algebraic varieties, which in turn are the two main pillars of classification theory of higher dimensional varieties. One of the main goals of the project is to further develop the theory of rational singularities in arbitrary characteristic. In particular, the investigator is working toward obtaining criteria which are easier to check than the current definition as well as toward establishing the existence of partial resolutions with rational singularities. Another of the main goals is to extend the definition of Du Bois singularities to arbitrary characteristics and study their deformation theoretic properties and their connections to rational singularities. Yet another goal of the project is to better understand the relatively new notion of 'liftable local cohomology' and its relations to deformations. The results obtained with regard to singularities and liftable local cohomology will be used in applications to moduli theory. 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.