### Practical

The session took place on Thursday **October 10** 2024 at the **University of Stuttgart**.

The address is:

room V 57.04 (ground floor)Pfaffenwaldring 57

70569 Stuttgart

Germany

### Directions from Stuttgart Hauptbahnhof (Main Station):

Stuttgart Hauptbahnhof is currently undergoing extensive construction. Follow the signs to the underground S-Bahn station, which is about a 5-minute walk away. Take any S-Bahn from lines S1, S2, or S3, and disembark at the Stuttgart Universität station. Important: After exiting the train, head toward the stairs at the rear of the platform.

Signs will be posted to guide participants to the Pfaffenwaldring 57 (PWR 57) building, one of the two large, twin concrete structures nearby.

URTAGS will take place on the ground floor, in room V 57.04.

### Schedule

10:45–12:15 | Vladimir Lazić: The Abundance conjecture |

12:15–13:30 | lunch |

13:30–14:30 | Simon Brandhorst: On the ramification indices of the period map of elliptic Enriques surfaces |

14:30–15:00 | coffee break |

15:00–16:00 | Lie Fu: Hyper-Kummer construction |

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

### Speakers, titles and abstracts

**Vladimir Lazić: The Abundance conjecture**

In this talk I will give a broad overview of the Abundance conjecture, which is one of the most important open problems in algebraic geometry. I will put it in the broader context of the classification of projective varieties, explain what is known and what remains to be proved, as well as elaborate on a recent strategy to use analytic techniques to prove the conjecture.

**Simon Brandhorst: On the ramification indices of the period map of elliptic Enriques surfaces**

(joint work with Víctor González-Alonso)

Barth and Peters showed that a generic complex Enriques surface has exactly 527 isomorphism classes Elliptic fibrations. We show that every Enriques surface has precisely 527 isomorphism classes of elliptic fibrations when counted with multiplicity. Their reducible singular fibers and the multiplicities can be calculated explicitly. The statements hold over any algebraically closed field of characteristic not two. To explain these results, we construct a moduli space of complex elliptic Enriques surfaces and study the ramification behavior of the forgetful map to the moduli space of unpolarized Enriques surfaces. Curiously, the ramification indices of a similar map compute the hyperbolic volume of the rational polyhedral fundamental domain appearing in the Morrison-Kawamata cone conjecture.

**Lie Fu: Hyper-Kummer construction**

Analogously to the classical construction of Kummer K3 surface from an abelian surface, S. Floccari discovered that any hyper-Kähler 6-fold of generalized Kummer type has a canonical action of a finite symplectic automorphism group such that the quotient admits a crepant resolution, which is a hyper-Kähler variety of K3[3]-type. In a joint work in progress with Floccari, we study various aspects of this “hyper-Kummer” construction. Our results include characterizations of the hyper-Kummer K3[3]-type variety in terms of lattices, relation between the derived categories and algebraic cycles (motives) of the hyper-Kummer 6-folds and the original generalized Kummer-type hyper-Kähler 6-folds. We especially highlight the rich geometry in this construction that many canonically associated hyper-Kähler 4-folds and K3 surfaces emerge and interact interestingly.