### Practical

The session took place on Tuesday **June 25** 2024 at the **University of Strasbourg**.

IRMA - Université de Strasbourg

Salle de conférence (ground floor)

10, rue du Général Zimmer

67000 Strasbourg, France

### Schedule

10:30–12:00 | Pieter Belmans: A panorama of quiver moduli |

12:00–13:30 | lunch (in the common room on the ground floor) |

13:30–14:30 | Ya Deng: On the universal covering of algebraic varieties |

14:30–15:00 | coffee break |

15:00–16:00 | Davide Cesare Veniani: Non-degeneracy of Enriques surfaces |

We intend to always feature one (longer) talk of a more survey-ish nature in each session.

### Speakers, titles and abstracts

**Pieter Belmans: A panorama of quiver moduli**

In this talk I will survey recent progress on the geometry of quiver moduli. These moduli spaces of (semi)stable quiver representations have many similarities with moduli spaces of vector bundles on curves, provided that you approach them from the right perspective. I will illustrate this dictionary between quivers and curves, in order to motivate how these two types of moduli spaces behave similarly from the point of view of symmetries, deformation theory, derived categories, stacky constructions, Brauer groups, rationality questions, etc. These similarities take on the form of analogies in the results, but also in the ingredients of the proofs.

**Davide Cesare Veniani: Non-degeneracy of Enriques surfaces**

Enriques’ original construction of Enriques surfaces dates back to 1896. It involves a 10-dimensional family of sextic surfaces in the projective space which are non-normal along the edges of a tetrahedron. An even earlier construction of Enriques surfaces is due to Reye and is known as Reye congruences.

In a series of joint works with G. Martin and G. Mezzedimi, we have now settled two questions: (1) Do all Enriques surfaces arise through Enriques’ construction? (2) Do all nodal Enriques surface arise as Reye congruences? In my talk, I will illustrate our results and review the main ideas involved in their proofs, with a particular emphasis on the concept of non-degeneracy.

**Ya Deng: On the universal covering of algebraic varieties**

In his famous book *Basic Algebraic Geometry*, Shafarevich asked whether the universal covering of a complex projective variety is homomorphically convex (the so-called Shafarevich conjecture). Substantial progress on this conjecture was achieved by Campana, Kollar, Napier, Katzarkov, Ramachandran, Eyssidieux etc in the 1990s. In 2012 Eyssidieux et al. proved the Shafarevich conjecture for smooth projective varieties whose fundamental groups are complex linear. In this talk, I will report some recent progress on the extension of this conjecture to cases where the variety is singular or the fundamental groups are subgroups of general linear groups in positive characteristic. If time permits, I will explain some applications in algebraic geometry. This is based on joint work with Yamanoi and Katzarkov.

This event was funded by the University of Strasbourg Institute for Advanced Study (USIAS)