Schedule and Abstracts

The seminars will be held in "Aula Lagrange" of the Mathematics Department at the University of Turin.

Titles and Abstracts:

Simon Brandhorst: Computation of the automorphism group of Enriques surfaces

(Joint work with Ichiro Shimada) The Morrison–Kawamata cone conjecture predicts that the action of the automorphism group of a Calabi-Yau variety on its effective nef cone admits a fundamental domain which is a rational polyhedral cone.

The conjecture is wide open in general. But Namikawa has verified it for Enriques surfaces. It follows that an Enriques surface admits up to the action of the automorphism group only finitely many smooth rational curves, elliptic fibrations, projective models of a given degree and its automorphism group is finitely generated and in fact finitely presented.

Naturally, enumerative questions arise:

- Can one explicitly describe a fundamental domain?

- How many smooth rational curves or elliptic fibrations are there up to the action of the automorphism group?

- Can one give generators for the automorphism group?

In this talk I will describe an algorithm to answer these questions.

Andreas Leopold Knutsen: Moduli spaces of polarized Enriques surfaces and their possible unirationality

Moduli spaces of polarized Enriques surfaces (parametrizing  isomorphism classes of pairs (S,H), where S is a smooth compact, complex Enriques surface, and H is an ample line bundle on S) have several components, even if one fixes the degree of the polarization. I will

present some new results showing how one can determine the various irreducible components in terms of intersection with suitable sequences of isotropic divisors, and how one can prove that infinitely many of these components are unirational.

In particular, there is a particular component dominating all others, and according to an announced (but yet unpublished) result of Gritsenko, this space should be of general type, showing that not all moduli spaces can be of negative Kodaira-dimension.

The talk will be based upon the recent works "On moduli spaces of polarized Enriques surfaces", J. Math. Pures Appl. 144 (2020) and "Irreducible unirational and uniruled components of moduli spaces of polarized Enriques surfaces" (with

C. Ciliberto, T. Dedieu, and C. Galati), Math. Z. 303 (2023).

Laura Pertusi: Moduli spaces of stable objects in Enriques categories

Enriques categories are characterized by the property that their Serre functor is the composition of an involutive autoequivalence and the shift by 2. The bounded derived category of an Enriques surface is an example of Enriques category. Other interesting examples are provided by the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids.

In this talk, we study moduli spaces of semistable objects in the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids with respect to Serre-invariant stability conditions. We provide a result of non-emptiness for these moduli spaces, by using the relation with certain moduli spaces on the associated K3 category. This is a joint work with Alex Perry and Xiaolei Zhao.

Yulieth Prieto-Montañez: K3 Surfaces and an Adversary of Enriques Surfaces

It is a well-established fact that complex Enriques surfaces can be represented as quotients of K3 surfaces through an involution group without fixed points. Drawing inspiration from this geometrical construction, which arises from considering cyclic coverings, we delve into the study of K3 surfaces as double covers of rational surfaces.

In this talk, the involutions under consideration are those that fix curves, differing from the Enriques construction. We explore elliptic fibrations of K3 surfaces induced by conic bundles of Del Pezzo surfaces. Our findings reveal the possibility of obtaining non-trivial elliptic fibrations on the K3 side. Following Kodaira's classification of singular fibers, we classify all elliptic fibrations on the K3 surfaces admitting these involutions. This classification can be derived through classes of conic bundles on the rational surfaces that align with the geometrical model, in accordance with the findings by Manin and Dolgachev.

This ongoing project is a collaboration with Pedro Montero, Paola Comparin, and Sergio Troncoso.

Sofia Tirabassi: Canonical Lift of Ordinary Enriques Surfaces

We introduce the notion of ordinary Enriques surface and we show that these admits a canonical lift. This is in collaboration with R. Laface.

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.

Short talks

Dario Faro (Università di Pavia): Gaussian maps and curves on Enriques surfaces

Let C be a complex projective algebraic curve and L and M be two line bundles on C.  One can associate L and M with some natural maps between spaces of global sections of certain sheaves on C. These are called Gaussian-Wahl maps. These maps have been classically studied in connection with extendability questions of curves on surfaces. In this talk I will focus on the case of Enriques surfaces, presenting some natural questions that arise in this situation.

Simone Pesatori (Università degli Studi Roma Tre): Enriques surfaces of base change type

We review a construction due to Hulek and Schütt of some particular Enriques surfaces: starting from a rational elliptic surface, they produce families of Enriques surfaces having an elliptic pencil with a (possibly singular) rational bisection which splits in two smooth curves in the k3 cover.
We investigate the geometry of these surfaces and relate their existence to the nonemptiness of some Severi varieties of P^2 and F_2.

Franco Giovenzana, Annalisa Grossi (Université Paris-Saclay): Enriques manifolds

Enriques manifolds were introduced by Boissière-Nieper-Wisskirchen-Sarti and Oguiso-Schröer as a generalisation in higher dimension of Enriques surfaces. In this talk we will revise the basics and the known examples of Enriques manifolds, then we will discuss a possible way to get new examples. Based on joint work in progress with S.Billi, and  L.Giovenzana.