From Enriques surface to Artin-Mumford counterexample

TL;DR

This paper explores the connection between Enriques surfaces and the Artin-Mumford counterexample to the Lüroth problem, providing a geometric construction linked to classical algebraic geometry.

Contribution

It offers an explicit construction of the Artin-Mumford counterexample based on the geometry of Enriques surfaces and Reye congruences, highlighting their relationship.

Findings

Explicit geometric construction of the Artin-Mumford example

Connection between Enriques surfaces and rationality counterexamples

Insights into classical algebraic geometry structures

Abstract

After an Introduction to the themes of Enriques surfaces and Rationality questions, the Artin-Mumford counterexample to Lueroth problem is revisited. A construction of it is given, which is related in an explicit way to the geometry of Enriques surfaces, more precisely to the special family of Reye congruences and their classical geometry.