Studies in Moduli Theory and Birational Geometry

点击翻译按钮获取中文摘要

The area of study of this project lies within algebraic geometry, thebranch of mathematics devoted to geometric shapes called algebraicvarieties, defined by polynomial equations. Algebraic geometryhas significant applications in theoretical physics, where physicists consideralgebraic varieties as a piece of the fine structure of ouruniverse. This is especially true with the first topic, modulitheory. This theory studies a remarkable phenomenon in which thecollection of all algebraic varieties of the same type is oftenmanifested as an algebraic variety, called a moduli space, in its ownright. Thus in algebraic geometry, the metaphor of thinking about acommunity of "organisms" as itself being an "organism" is not just ametaphor but a rigorous and quite useful fact. The other topic studied inthis project is birational geometry, focusing here on resolution of singularities, applied in this project to families of varieties. Resolution of singularities is a fundamental procedure where "bad" points of an algebraic variety are removed and replaced by "good" points.Abramovich will continue studying problems in birational geometry, specifically the problem of functorial semistable reduction. Here Abramovich and collaborators will build on Hironaka's method of resolution of singularities in order to resolve singularities of families of varieties. Long term goals include extending this effort to other geometric categories and to singular foliations. In addition, Abramovich will continue to study moduli spaces. The main foci are Moduli and arithmetic of K3 surfaces, where one wishes to find situations where K3 moduli spaces are algebraically hyperbolic; representability of logarithmic moduli, where an analogue of Artin's criteria is sought; a quest to describe explicit moduli of certain stable surfaces; and completion of a long-term project on the logarithmic degeneration formula.--------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.