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

TL;DR

This paper provides a comprehensive overview of rational and non-rational algebraic varieties, including key theorems, examples, and methods for constructing non-rational hypersurfaces, based on Kollár's lecture course.

Contribution

It offers a detailed exposition of rationality concepts, classical theorems, and new construction techniques for non-rational varieties in algebraic geometry.

Findings

Segre's theorem on non-rationality of certain cubic surfaces

Manin's theorem on projective equivalence of cubic surfaces

Kollár's method for constructing low degree non-rational hypersurfaces

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.