Studies in Moduli Theory and Birational Geometry

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The area of study of this project lies within algebraic geometry, the branch of mathematics devoted to geometric shapes called algebraic varieties, defined by polynomial equations. Algebraic geometry has significant applications in coding, industrial control, computation, and theoretical physics, where physicists consider algebraic varieties as a piece of the fine structure of our universe. One focus in this project is Moduli theory, which studies a remarkable phenomenon in which the collection of all algebraic varieties of the same type is often manifested as an algebraic variety in its own right, called a moduli space. Thus in algebraic geometry, the metaphor of thinking about a community of "organisms" as itself being an "organism" is not just a metaphor but a rigorous and quite useful fact. The other focus in this project is birational geometry, focusing here on resolution of singularities. Resolution of singularities is a fundamental procedure where "bad" points of an algebraic variety are removed and replaced by "good" points. This project includes research opportunities for undergraduate and graduate students.In more detail, the PI will continue studying problems in birational geometry, focusing on resolution of singularities, semistable reduction, and the geometry of stack theoretic birational modifications. The PI and collaborators will build on recently completed work to functorially resolve singularities of proper families of varieties; to study the geometry of weighted blowings up and more general stack-theoretic procedures; and to study subtle phenomena in positive characteristics. In addition, the PI will continue to study moduli spaces. The main foci are Moduli and arithmetic of subvarieties of families of abelian varieties, aiming to extend recent non-degeneracy and uniformity results of rational points on curves to symmetric squares of curves and related constructions.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.