Rational Points on Algebraic Varieties and Geometry of Curves and Surfaces

University Of Chicago, Chicago IL

Investigators

Abstract

This project addresses several fundamental questions in algebraic geometry and number theory. The first part concerns the geometry of solutions to algebraic equations. Let X be a variety defined over a number field K. Does there exist a finite extension L/K such that X(L) is dense? One seeks criteria for when this occurs in terms of the geometry of X. It is still not clear what form such criteria should take. The second part concerns the geometry of plane curves. When is an abstract curve a degeneration of smooth plane curves? What geometric properties do such limiting plane curves have in common? The third part involves the classification of surface singularities. Given a family of projective normal surfaces, when are the invariants of such a family constant? Specifically, how does one tell whether the self-intersection of the canonical divisor is constant? The last part involves finding explicit parametrizations for the curves of a fixed small genus. Algebraic geometers study the structure of solutions to polynomial equations. From ancient times, geometric figures like circles and ellipses have fascinated architects, scientists, and artists. The most compact way to represent such a figure is to describe it as the solutions to an equation. One basic question is to classify the abstract figures that may be represented by equations of a given form. Another is to understand the common geometric properties of the solutions to all the equations of a given form. This project seeks to answer these questions for special types of curves and surfaces.