Algebraic Geometry Genova-Nice-Torino meeting home
Schedule:
Monday 19/06 - Aula 509 DIMA
14:15 - 15:15 Valeria Bertini
15:15 - 15:50 Coffee break
15:50 - 16:50 Emanuele Ventura
17:00 - 18:00 Bruno Dewer
Tuesday 20/06 - Aula 508 DIMA
8:45 - 9:45 Sorin Dumitrescu
9:45 - 10:15 Coffee break
10:15 - 11:15 Riccardo Moschetti
11:25 - 12:25 Matteo Penegini
Titles and abstracts:
Valeria Bertini (Università di Genova)
Terminalization of quotients of hyperkähler manifolds via symplectic actions
A very fruitful way to produce terminal irreducible symplectic varieties is to start from a smooth hyperkähler manifold admitting a symplectic action on it, and to consider the terminalization of its quotient. In this talk I will start from hyperkähler manifolds that are deformation of Hilbert schemes or generalized Kummer varieties, and I will consider all possible examples obtained as above by acting on them with automorphisms induced by the underlying surface, for which a systematic analysis is possible. This is the content of a joint work with Armando Capasso, Olivier Debarre, Annalisa Grossi, Mirko Mauri and Enrica Mazzon.
Bruno Dewer (Université Paul Sabatier)
Gorenstein weighted projective spaces of dimension 3, maximal extensions and linear sections
Given a projective variety $X\subset \mathbf P^N$, it is said to be extendable if there exists $Y\subset \mathbf P^{N+r}$ which is not a cone and such that $X$ is a linear section of $Y$. This talk focuses on the list of the weighted projective spaces of dimension 3 which are Gorenstein and their extendability in their anticanonical model. Next, we will see a characterization of the primitive curves in their hyperplane sections.
Sorin Dumitrescu (Université Côte d'Azur)
Holomorphic sl(2, C)–differential systems on compact Rieman surfaces and curves in compact quotients of SL(2, C)
We explain the strategy of a recent result that constructs holomorphic sl(2, C)–differential systems over some Riemann surfaces Σg of genus g ≥ 2, such that the image of the associated monodromy homomorphism is some cocompact Kleinian subgroup Γ ⊂ SL(2, C). As a consequence, there exist holomorphic maps from Σg to the quotient SL(2, C)/Γ, that do not factor through any elliptic curve. This answers positively a question asked by Huckleberry and Winkelmann, also raised by Ghys.
Riccardo Moschetti (Università di Torino):
The combinatorial non-degeneracy invariant of Enriques surfaces
The geometry and combinatorics of elliptic fibrations on Enriques surfaces is fairly well understood for general Enriques surfaces (i.e., Enriques surfaces without smooth rational curves, also called unnodal). However, the behavior of elliptic fibrations changes radically when we specialize our Enriques surface. The non-degeneracy invariant is the maximum number of half-fibers pairwise intersecting giving 1, where a half-fiber is a curve F on the surface such that |2F| is an elliptic pencil. This invariant encodes many relations about the geometry of an Enriques surfaces, and of its derived category. I will talk about a series of ongoing works joint with Franco Rota and Luca Schaffler, concerning the computation of this invariant in some non-general cases.
Matteo Penegini (Università di Genova)
Hodge structures of K3 type of bidouble covers of rational surfaces
A bidouble cover is a flat G := (Z/2Z)^2-Galois cover X → Y . In this situation there exist three intermediate quotients Y_1, Y_2 and Y_3 which correspond to the three subgroups Z/2Z ≤ G. In this talk we will consider the following situation: Y will be a rational surface and Y_i will be either a surface with p_g = 0 or a K3 surface. These assumptions will enable us to have a strong control on the weight 2 Hodge structure of the covering surface X. In particular, we classify all covers with these properties if Y is minimal, obtaining surfaces X with p_g(X) = 1,2,3. Moreover, we will discuss the Infinitesimal Torelli Property, the Chow groups and Chow motive, and the Tate and Mumford-Tate conjectures for X. We also introduce another construction, called iterated bidouble cover, which allows us to obtain surfaces with higher value of p_g for which we still have a strong control on the weight 2 Hodge structure.
Emanuele Ventura (Politecnico di Torino)
Tensor rank: geometry and complexity theory
Tensor rank gives a measure for the arithmetic complexity of computing a bilinear map such as matrix multiplication. In the last decade, there has been intense activity to find algebraic geometry approaches to bound tensor ranks. I will describe a recent result of Buczyńska and Buczyński leading to the notion of border varieties of sums of powers. These are subvarieties of multigraded Hilbert schemes and I will discuss some examples of border varieties for plane curves. This is a joint work in progress with Tomasz Mańdziuk.
Algebraic Geometry Genova-Nice-Torino meeting home