Filippo Favale (University of Pavia)

Title: Hessian varieties and cubic hypersurfaces

Abstract: If $X=V(f)$ is a hypersurface of $\mathbb{P}^n$ of degree $d$, one can construct the Hessian matrix $H_f$, a matrix whose entries are homogeneous forms of degree $d-2$, and use it to construct a filtration of $\mathbb{P}^n$ with respect to the rank of $H_f$. The ``highest'' stratum is the Hessian variety $\mathcal{H}_f$, i.e., the locus of points of $\mathbb{P}^n$ where $H_f$ is not of maximal rank. This is the zero locus of the hessian (polynomial) associated to $f$, namely $h_f=\det(H_f)$.

Hessian hypersurfaces exhibit a rich geometry, which becomes even more abundant in the specific case of cubics, the case on which we will focus for this talk. We will mainly discuss two results, in collaboration with G.P. Pirola and D. Bricalli, regarding the geometry of the singular locus of the Hessian variety of a general cubic fourfold, and the problem of characterizing the smooth cubic threefolds and fourfolds for which the Hessian variety is reducible.

Matteo Fiacchi (University of Ljubljana)

Title: On the Minimal Metric in Real Euclidean Spaces

Abstract: Recently, Forstnerič and Kalaj introduced a metric, called the minimal metric, in domains of $\mathbb{R}^d$, which is defined using conformal minimal disks. This metric is analogous, in the theory of minimal surfaces, of the Kobayashi metric in domains of $\mathbb{C}^d$ , which is defined using holomorphic disks.

In this seminar, we will give an overview of this metric and discuss its metric geometry properties. Finally, we will focus on the relationship between the minimal metric and the Hilbert metric in convex domains of  $\mathbb{R}^d$.

Tobias Harz (University of Berna)

Title: On plurisubharmonic defining functions for pseudoconvex domains in C^2

Abstract: We investigate the question of existence of plurisubharmonic defining functions for smoothly bounded, pseudoconvex domains in C2. In particular, we construct a family of simple counterexamples to the existence of plurisubharmonic smooth local defining functions. Moreover, we give general criteria equivalent to the existence of plurisubharmonic smooth defining functions on or near the boundary of the domain. These equivalent characterizations are then explored for some classes of domains. This is joint work with A.-K. Gallagher. 

Hans-Joachim Hein (University of Münster)

Title: A gluing construction for complex surfaces with hyperbolic cusps

Abstract: We will describe an example of a degeneration of degree 6 algebraic surfaces in CP^3 with an isolated triple point singularity on its central fiber. Then we will show how the unique negative Kähler-Einstein metrics on the smooth fibers, which exist by the Aubin-Yau theorem, disintegrate into three distinct geometric pieces on approach to the central fiber: (1) Kobayashi's complete Kähler-Einstein metric on the complement of the triple point, (2) long thin neck regions, and (3) Tian-Yau's complete Ricci-flat Kähler metric in small neighborhoods of the vanishing cycles. Joint work with Xin Fu and Xumin Jiang. 

Marc Levine (University of Duisburg-Essen)

Title: Quadratic Euler characteristics and symmetric powers

Abstract: Symmetric powers for a commutative ring R can be defined using a so-called power structure for R, and under some reasonable assumptions on R, the two structures are equivalent. For the Grothendieck ring of varieties over a field, K_0(Var_k),  sending a smooth projective variety X to its nth symmetic power Sym^n(X) extends to a power structure, as was shown by Gusein-Zade, Luengo and Melle-Hernández.  Recently, Pajwani and Pál have defined a power structure on the Grothendieck-Witt ring of non-degenerate quadratic forms over a field k, GW(k). There has been a number of results describing a possible compatibility between these two notions of symmetric powers, using the motivic measure defined by the compactly supported quadratic Euler characteristic of Arcila-Maya, Bethea, Opie, Wickelgren and Zakharevich.  My talk will describe some of these recent results and indicate possible applications. 

Barbara Nelli (University of L'Aquila)

Title:  Minimal Graphs with infinite boundary values in 3-manifolds 

Abstract: In the sixties, H. Jenkins and J. Serrin proved a famous theorem about minimal graphs in the Euclidean 3-space with infinite boundary values. After reviewing the classical results, we show how to solve the  Jenkins-Serrin problem in a 3-manifold with a Killing vector field.  This is a joint work with A. Del Prete and J. M. Manzano.

Carla Novelli (University of Torino)

Title: Triple solids and scrolls

Abstract: Let Y be a smooth projective variety of dimension n ≥ 2 with a finite morphism Y -> P^n of degree 3. Suppose that Y, polarized by some ample line bundle, is a scroll over a smooth variety X of dimension m. Then n≤3, hence m=1 or 2. When m=1, a complete description of the few varieties Y that satisfy these conditions will be given. When m=2, various restrictions will be discussed showing that in several instances the possibilities for such a Y reduce to the single case of the Segre product P^2 x P^1. This is a joint work in collaboration with Antonio Lanteri.

Luca Schaffler (University of Rome 3)

Title: The non-degeneracy invariant of Enriques surfaces: a computational approach

Abstract:  For an Enriques surface S, the non-degeneracy invariant nd(S) retains information about the elliptic fibrations of S and its projective realizations. While this invariant is well understood for general Enriques surfaces, computing it becomes challenging when specializing our Enriques surface. In this talk, we introduce a combinatorial version of the non-degeneracy invariant that depends on S along with a configuration of smooth rational curves, and gives a lower bound for nd(S). We also provide a SageMath code that computes this combinatorial invariant and we apply it in several examples where nd(S) was previously unknown. In particular, we study the families of Enriques surfaces introduced by Brandhorst and Shimada. The results presented are joint works and ongoing projects with Riccardo Moschetti and Franco Rota. 

Andrea Tamburelli (University of Pisa)

Title: Bi-complex hyperbolic space and SL(3,C)-quasi-Fuchsian representations

Abstract: The bi-complex hyperbolic space is a homogeneous space with negative constant para-holomorphic sectional curvature. After describing its main features, we will define minimal Lagrangian surfaces in this space and their structural equations. If they are equivariant under the action of a representation of a surface group into SL(3,C), we will see that their embedding data provide a parameterization of the space of SL(3,C)-quasi-Fuchsian representations by two copies of the bundle of holomorphic cubic differentials over Teichmüller space. 

Michela Zedda (University of Parma)

Title: Projectively induced Kaehler metrics and related issues

Abstract:  This talk gives an overview of the problematic related to projectively induced and balanced Kaehler metrics, and to the existence of a Tian-Yau-Zelditch expansion of the epsilon-function. Differences between compact and noncompact case will be highlighted. The talk is based on joint works with Simone Cristofori (Università di Parma), Andrea Loi and Fabio Zuddas.