Moduli problems and root constructions in algebraic geometry



Andrew Kobin
University of Virginia

Thursday, March 28, 2019
Manchester Hall, Room 018

Moduli problems and root constructions in algebraic geometry

Many classification problems in math – usually stated as “describe all isomorphism classes of (blank)”, or “how many (blanks) are there?” – have a natural parametrization or geometric structure to them. A moduli problem is, loosely speaking, such a classification problem with natural geometric structure. In the first part of this talk, I will interpret several problems in geometry as moduli problems and show how algebraic geometry can be used to solve them. Among these are: the number of lines on a cubic surface (spoiler: there are exactly 27) and the theory of modular forms. In the second part, we will discuss how to take n-th roots — not just of numbers, but of more general objects in math, and in particular of line bundles. Then, I will explain how modular forms over the complex numbers relate to the problem of taking nth roots of certain line bundles over a curve. Time permitting, I will introduce a new construction that I am using to shed light on these ideas in characteristic p.   


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