Rational and Non-Rational Algebraic Varieties: Lectures of J\'anos Koll\'ar

Abstract

This is a detailed write-up of Koll\'ar's course at the EMS summer school in Algebraic Geometry in Eger, Hungary, August 1996. The topics include definitions and examples of rational and unirational varieties, with special attention to varieties defined over non-algebraically closed fields, Segre's theorem on non-rationality of cubic surfaces of Picard number one, Manin's theorem that birationally equivalent cubic surfaces of Picard number one are projectively equivalent, and Koll\'ar's method for constructing examples of low degree non-rational hypersurfaces. Includes many exercises and their solutions.