From Enriques surface to Artin-Mumford counterexample

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.Comment: To appear in 'Algebraic Geometry between Tradition and Future', Proceedings of the INdAM Workshop 'Italian algebraic geometry between tradition and future', Rome December 6-8 2021. Springer INdAM Series. 23 page