A primer on measures of irrationality

Abstract

Measures of irrationality are a numerical way of quantifying how far a given variety is from being rational (or rationally connected, uniruled, etc.). In the last two decades, there has been renewed interest in the study of these invariants. The goal of this expository survey is to summarize known results from the point of view of the Kodaira--Enriques classification of surfaces, highlight recent progress, and discuss a number of open problems and questions.

arXiv comment29 pages; this work grew out of the March 2024 AIM workshop "Degree d points on algebraic surfaces"