Geometry of Rationally Connected Varieties

Harvard University, Cambridge MA

Investigators

Abstract

Rationally connected varieties form a class of varieties that coincides with rational and unirational varieties for curves and surfaces, but represents a very different (and better behaved) class in higher dimensions. The investigator studies aspects of rational connectivity; in particular, the relation between rational connectivity and unirationality and the implications of rational connectivity for existence of points on varieties over non-algebraically closed fields. In relation to the latter, the investigator studies the geometry of parameter spaces for rational curves on a rationally connected variety. The central goal of algebraic geometry is to learn more about the solutions of polynomial equations by studying geometric objects, called varieties, associated to them. For example, if an ellipse is given to us by a polynomial in two variables, we might ask how the shape of an ellipse affects the solutions of the polynomial. Recently, algebraic geometers have introduced a new property of varieties, called rational connectivity, and results to far indicate a strong connection between the algebra of a system of polynomial equations and the rational connectivity of the variety they define. The investigator studies this connection further.