Tensor eigenvectors are a generalization of matrix eigenvectors, and their definition induces a natural structure of standard determinantal scheme, which is called eigenscheme. We shall illustrate some characterization of their geometry by using projective bundle techniques, Gruson - Peskine numerical character and classical liaison techniques. We will finally apply some of these results to Jacobian schemes of conic-line arrangements with high total Tjurina number.

We will talk about a structure result for some (smooth, complex) Fano varieties X, which depends on the Lefschetz defect delta(X), an invariant of X defined as follows. Consider a prime divisor D in X and the restriction r:H^2(X,R)->H^2(D,R). Then delta(X) is the maximal dimension of ker(r), where D varies among all prime divisors in X. If delta(X)>3, then X is isomorphic to a product SxT, where S is a surface. When delta(X)=3, X does not need to be a product, but we will see that it still has a very explicit structure. More precisely, there exists a smooth Fano variety T with dim T=dim X-2 such that X is obtained from T with two possible explicit constructions; in both cases there is a P^2-bundle Z over T such that X is the blow-up of Z along three pairwise disjoint smooth, irreducible, codimension 2 subvarieties. This structure theorem allows to complete the classification of Fano 4-folds with Lefschetz defect at least 3. This is a joint work with Eleonora Romano and Saverio Secci.

In this survey talk, I will discuss the problem of which Hodge classes on abelian varieties are represented by subvarieties, smooth or not, cycles, or sums of products of Chern classes of vector bundles.

In this talk, I will discuss projective log smooth pairs with numerically flat normalized logarithmic tangent bundle. Generalizing works of Jahnke-Radloff and Greb-Kebekus-Peternell, we show that, passing to an appropriate finite cover and up to isomorphism, these are the projective spaces or the log smooth pairs with numerically flat logarithmic tangent bundles blown-up at finitely many points away from the boundary. On the other hand, the structure of log smooth pairs with numerically flat logarithmic tangent bundle is well understood: they are toric fiber bundles over finite etale quotients of abelian varieties.

My talk is devoted to an old theorem of Borho and MacPherson which says that the affine cone of nilpotent elements in the lie algebra of  a complex algebraic group is rationally smooth. After reviewing Borho-MacPherson's proof, I focus on a recent approach, due to a collaboration with V. Di Gennaro and C. Sessa, which allows us to prove a slightly more general result.

We will present a study of  the local geometry of the moduli space of intermediate Jacobians of (2,2)-threefolds in P^2 x P^2. More precisely, we will prove that a composition of the second fundamental form of the Siegel metric in A_9 restricted to this moduli space, with a natural multiplication map, is a nonzero holomorphic section of a vector bundle. We will also give an explicit description of its kernel. This is a joint work with Elisabetta Colombo, Juan Carlos Naranjo and Gian Pietro Pirola.

We are all familiar with eigenvalues of matrices and their importance, both in mathematics and in the applied sciences. When we pass from matrices to tensors, the situation becomes more complicated. In this talk I'll introduce the characteristic polynomial of a tensor and I'll discuss some open question. In particular, I'll focus the identifiability of a tensor from its eigenvalues.

In algebraic geometry, understanding the geometry of moduli spaces is a beautiful and powerful way to understand the classes of objects that they parametrize.
In the last few years, it has been understood that these moduli spaces can often be “tropicalized” via modular maps which allow to study many properties of the original spaces by looking at their tropical counterpart. In the talk, I will try to explain this interplay in the case of the moduli space of curves, (universal) Jacobians and, time permitting, the moduli space of abelian varieties.

I will show an inequality of Brill-Noether flavour on the Euler characteristic of coherent sheaves on complex abelian varieties, and an application to singularities of divisors of low degree.

The talk is based on our recent arXiv preprint with Giovanni Staglianò on the subject.
We shall first present some general results on rational complete intersections of dimension 4, on their associated congruences and on their associated surfaces.
Then we shall construct explicit birational maps from some rational complete intersection of three quadrics in P^7 to some prime Fano manifolds together with their Sarkisov decomposition via a single Secant Flop, allowing us to recover the cohomologically associated Castelnuovo surface of general type with K^2 = 2 and χ = 4 (the double cover of P^2 ramified along the discriminant curve of the net of quadrics defining the complete intersection) as the minimal model of the non ruled irreducible component of the base locus of the inverse maps.

Enriques manifolds are non simply connected manifolds whose universal cover is irreducible holomorphic symplectic, and as such they are natural generalizations of Enriques surfaces. The goal of the talk is to prove the Morrison-Kawamata cone conjecture for such manifolds when the degree of the cover is prime using the analogous result (established by Amerik-Verbitsky) for their universal cover. If time permits I will also show the cone conjecture for the known examples having non-prime degree. This is a joint work with Gianluca Pacienza.

Rüdiger Achilles
Alberto Alzati
Edoardo Ballico
Lorenzo Barban
Valentina Beorchia
Cinzia Bisi
Claudio Braggio
Maria Chiara Brambilla
Davide Bricalli
Alberto Calabri
Stefano Canino
Cinzia Casagrande
Alex Casarotti
Federico Caucci
Priyankur Chaudhuri
Elisabetta Colombo
Olivier Debarre
Francesco Antonio Denisi
Stéphane Druel
Philippe Ellia
Filippo Francesco Favale
Davide Franco
Paola Frediani
Francesco Galuppi
Alessandro Gimigliano
Gianluca Grassi
Annalisa Grossi
Monica Idà
Remke Kloosterman
Mirella Manaresi
Alex Massarenti
Massimiliano Mella
Margarida Melo
William Montoya Catano
Gianluca Occhetta
Giorgio Ottaviani
Giuseppe Pareschi
Alessandro Passantino
Andrea Petracci
Elena Poma
Dmitrii Rachenkov
Francesco Russo
Pierpaola Santarsiero
Alessandra Sarti
Enrico Schlesinger
Luis Eduardo Solà Conde
Paola Supino
Orsola Tommasi