Degenerations and moduli

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Dr. Alexeev will work on a range of problems in algebraic geometry centering around degenerations, compact moduli spaces of stable varieties and pairs, and connections to the Minimal Model Program. Stable varieties and pairs are higher-dimensional generalizations of Deligne-Mumford-Knudsen's stable curves. Some particular subjects of this project are moduli of weighted stable hyperplane arrangements, stable surfaces of general type, and geometric meaning of various compactifications of moduli spaces of abelian varieties and K3 surfaces.Algebraic geometry is one of most central branches of mathematics which aims to understand, both practically and conceptually, solutions of systems of polynomial equations in many variables. It has important applications to other fields of mathematics, such as number theory, topology, analysis, as well as to physics, biology, cryptography, and engineering. The particular problems that Dr Alexeev will study involve moduli spaces, a classical subject going back to Riemann, and in particular moduli spaces of stable curves and maps, and their higher-dimensional analogues. Some of these originated in physics, and may in turn find application there. The grant will contribute to training of graduate students, and will support an active program in algebraic geometry at the University of Georgia.