Valery Alexeev will continue his work on a wide variety of projects in algebraic geometry, many of them revolving around the construction and study of the moduli spaces of stable pairs and maps, the higher-dimensional generalizations of Deligne-Mumford-Knudson-Kontsevich's moduli spaces of stable curves and maps, as well as questions related to the Minimal Model Program, and to varieties with group action. The grant will contribute to training of graduate students and postdocs, and will support an active program in algebraic geometry at the University of Georgia.
Algebraic geometry arose in the ancient times from the study of polynomial equations. It now employs a dazzling array of sophisticated tools and methods, and has connections and applications to most other fields of mathematics, as well as to physics and engineering. The study of moduli spaces is at the heart of algebraic geometry and answers questions such as: What is the totality of algebraic objects of a given type? Does it have a special structure? What happens to the objects when they "degenerate", for example when they are stretched to their physical limits?
The aim of the project was to investigate a number of problems in algebraic geometry, which is a major part of mathematics dealing with solutions of systems of polynomial equations, using a variety of algebraic and geometric methods. Algebraic geometry is of fundamental importance in mathematics and its application in science and engineering. More specifically, the aim of the project was to investigate degenerations of algebraic varieties and the compact moduli of stable varieties and stable pairs. These subjects have important application in several fields of mathematics, such as arithmetic geometry, symplectic geometry, mirror symmetry. They also have applications in mathematical physics. The main outcome of the project is publication of seven refereed research papers, including papers in such leading mathematical journals as Inventiones Mathematicae, Compositio Mathematica, Journal of Algebra, and Crelle’s Journal (Journal fur die reine und andewandte Mathematik). The grant also supported four graduate students working with the principal investigator, and the work of the wider algebraic geometry group at the University of Georgia, speakers in the seminar and visitors. In the Inventiones paper, Alexeev and Brunyate disprove a 1973 conjecture about the regularity of the extended Torelli map to the Igusa compactification of the moduli space of principally polarized abelian varieties of dimension >8, and prove the regularity of the extended map to the perfect cone compactification. This topic is further carried on in another paper dedicated to dimensions 6,7,8. In the Compositio paper, Alexeev and Pardini extend the theory of abelian Galois covers to the case of singular varieties, which appear on the boundary of moduli spaces of varieties of general type. In the Journal of Algebra, Alexeev and Hacon investigate the depth properties of log canonical singularities. In two papers, Alexeev, together with Swinarski and with Gibney-Swinarski investigate special line bundles on the moduli spaces of stable curves, such as GIT bundles and conformal block bundles.
