This project addresses the geometry of moduli spaces that parameterize algebraic and geometric objects. The focus will be on spaces where there are several natural approaches to the moduli problem, particularly via geometric invariant theory, moduli of pairs, and Hodge theory. The former two approaches provide a method where the geometry of the objects parameterized plays a central role. The latter approach can allow for the use of arithmetic methods, including modular forms. Specific problems to be considered include giving modular interpretations to boundary loci of arithmetic quotients, describing log-canonical models of the moduli space of curves, and investigating the local structure of compactified Jacobians. Abelian varieties and theta divisors will also be of special interest.
Algebraic geometry is the field of mathematics that focuses on the solution sets of polynomial equations. Inasmuch as polynomial equations are ubiquitous, the subject sits at the crossroads of many different fields including complex geometry, number theory and theoretical physics. A motivating question for algebraic geometers has been to classify solution sets by their invariants. For instance, one could try to classify those complex valued solution sets that can be identified with a torus (the surface of a donut); this would correspond to fixing the invariants known as the (complex) dimension and genus equal to the number one. Often the collection of solution sets with fixed invariants can itself be viewed naturally as a solution set of polynomial equations. These are known as moduli spaces, and their algebraic and geometric properties yield a tremendous amount of information about the original solution sets of interest. The PI intends to study a number of such spaces. The proposed research will have a significant impact on a field that plays a central role in mathematics and interacts with numerous other fields.
Algebraic geometry is the field of mathematics that focuses on the solution sets of polynomial equations. Inasmuch as polynomial equations are ubiquitous, the subject sits at the crossroads of many different fields including complex geometry, number theory and theoretical physics. A motivating question for algebraic geometers has been to classify solution sets by their invariants. For instance, one could try to classify those complex valued solution sets that can be identified with a torus (the surface of a donut); this would correspond to fixing the invariants known as the (complex) dimension and genus equal to the number one. Often the collection of solution sets with fixed invariants can itself be viewed naturally as a solution set of polynomial equations. These are known as moduli spaces, and their algebraic and geometric properties yield a tremendous amount of information about the original solution sets of interest. The PI studied a number of such spaces. The results will have a significant impact on a field that plays a central role in mathematics and interacts with numerous other fields. Specifically, this project addressed the geometry of moduli spaces that parameterize algebraic and geometric objects. The focus was on spaces where there are several natural approaches to the moduli problem, particularly via geometric invariant theory, moduli of pairs, and Hodge theory. The former two approaches provide a method where the geometry of the objects parameterized plays a central role. The latter approach can allow for the use of arithmetic methods, including modular forms. Specific problems considered included giving modular interpretations to boundary loci of arithmetic quotients, describing log-canonical models of the moduli space of curves, and investigating the local structure of compactified Jacobians. Abelian varieties and theta divisors were also of special interest. Significant results were achieved in all of these directions, with the results published in scholarly journals, and made available on the pre-print archive, arXiv.org. The projects performed by the PI lie at the intersection of several fields which continue to attract the interest of a great number of algebraic geometers. The birational geometry of moduli spaces has been an important and central area of research for many years, and has numerous connections with the study of abelian varieties. The theory of algebraic curves is one area where this connection is most evident, and the work of the PI explored this further through study of the geometry of degenerate theta divisors. Arithmetic quotients are of interest in a number of settings, and modular interpretations of their compactifications are of broad interest. The results of the project have furthered research in these areas. The PI has played an active role in the mathematics community with the intent of disseminating the results of this project, contributing to the advancement of discovery, and promoting learning at all levels. The PI continues to be involved in organizing the Front Range Algebra, Geometry and Number Theory Seminar, and the Western Algebraic Geometry Symposium, both of which bring researchers from a wide geographic area together to discuss their work, and enhance partnerships among institutions. The PI has given lectures on the results of this project at institutions around the world. The PI has been committed to encouraging the participation of underrepresented groups in mathematics.
