All the conferences will take place in the Lecture Hall at the ground floor.

Click here to download the program, with titles and abstracts of the talks.

At the end of the workshop there will be a drinks party offered to the every partecipant.

Schedule

9.30: A method for the computation of the cohomologies of the restricted tangent and the normal bundle on smooth rational curves

Alberto Alzati (Università degli Studi di Milano)

Ogni curva razionale liscia di grado d è la proiezione della d-immersione di Veronese di P^1 = P(U) da un opportuno spazio P(T); T può essere considerato come un sottospazio vettoriale di S^d(U). Il metodo in questione si basa sulle proprietà di T rispetto ad alcuni operatori GL(U)-invarianti. Nella conferenza verrà brevemente illustrato tale metodo e alcune sue applicazioni (curve monomiali, spazi dei moduli, algoritmi combinatori, ecc.) nonché possibili generalizzazioni alle varietà razionali.

I risultati sono contenuti in una serie di lavori svolti in collaborazione con R. Re ed A. Tortora per le applicazioni combinatorie.

10.30: Coffee Break

11.00: Calabi-Yau 4-folds of Borcea-Voisin type with many fibrations

Alice Garbagnati (Università degli Studi di Milano)

The Calabi--Yau varieties are a special class of varieties with trivial canonical bundle. They are interesting both in algebraic geometry, because of the absence of a canonical model, and in physics, as they are crucial in the study of the mirror symmetry and the string theory.

A way to construct Calabi-Yau varieties is to apply the so called Borcea-Voisin construction, which produces a Calabi--Yau variety $X$ of dimension $n=n_1+n_2$ starting from two Calabi--Yau varieties $Y_1$ and $Y_2$ of dimension $n_1$ and $n_2$ respectively.

The aim of this talk is to illustrate the Bocea--Voisin construction and to apply it to specific choices of Calabi-Yau surfaces $S_1$ and $S_2$. This allows to construct Calabi--Yau 4-folds $X$ which admit several fibrations; in particular $X$ admits at least one fibration whose general fibers are $m$-dimensional Calabi--Yau varieties for any $m$ lower than 4. One of the fibration in curves, which is an elliptic fibration, has fibers of type I_5 on a del Pezzo surface which is contained in the basis of the fibration. This property seems interesting for the F-theory.

This is a joint work with A. Cattaneo and M. Penegini.

12.00: Supergeometry, Non Projected Calabi-Yau Supermanifolds and their Embedding

Simone Noja (Università degli Studi di Milano)

La supergeometria è lo studio di varietà caratterizzate da fasci di algebre Z_2-graduate i cui elementi pari commutano ed i cui elementi dispari anticommutano. Nella prima parte del talk verranno introdotte le premesse matematiche e le motivazioni fisiche che hanno condotto all’introduzione di questo tipo di geometrie. Nella seconda parte verranno presentati alcuni risultati recenti relativamente a supervarietà di Calabi-Yau su P^2 “non-projected”, la cui geometria, cioè, non può essere ricostruita a partire dall’ordinaria varietà sottostante. In particolare, verranno indagate questioni relative all’immersione di questa classe di supervarietà: si mostrerà che, sebbene tali varietà non ammettano immersioni in super spazi proiettivi - non sono cioè, superproiettive -, esse possono sempre essere immerse in super Grassmanniane.

13.00: Lunch Break

14.30: On 3-folds of general type with p_g = 1, 2 and 3

Matteo Penegini (Università degli Studi di Genova)

15.30: Coffee Break

In this talk I present a characterization for the birationality of the $5$--canonical map of a minimal algebraic threefold of general type with $p_g \geq 3$. Our result is an analogue of the characterization of the pluricanonical maps for surfaces of general type due to Bombieri. Moreover I will discuss similar results for threefolds of general type with p_g = 1 and 2. This is a jointwork with Meng Chen and Yong Hu.

16.00: Calabi-Yau quotients of irreducible holomorphic symplectic manifolds

Chiara Camere (Università degli Studi di Milano)

In this talk I will explain how to construct Calabi--Yau manifolds starting from irreducible holomorphic symplectic manifolds endowed with non-symplectic automorphisms of prime order. Then I will restrict to dimension four and describe the geometry of the Calabi--Yau fourfolds obtained as resolutions of quotients of fourfolds of K3^[2]-type by a non-symplectic involution, and finally use this to construct projective models of some fourfolds of K3^[2]-type.

This is joint work with Alice Garbagnati and Giovanni Mongardi.

17.00: On Shimura subvarieties of A_g contained in the Prym locus

Paola Frediani (Università degli Studi di Pavia)

I will present some results obtained in collaboration with E. Colombo, A. Ghigi and M. Penegini on Shimura subvarieties of A_g generically contained in the Prym locus. I will explain the construction of 1-dimensional families of double covers compatible with a fixed group action on the base curve C such that the quotient of C by the group is the projective line. I will give a simple criterion for the image of these families under the Prym map to be a Shimura curve. I will show that this criterion allows us to construct several examples of Shimura curves generically contained in the Prym locus in A_g for g < 13.

18.00: Drinks party