Schedule and Abstracts

The event will start on Monday at 14:00 in Sala Emma Strada on the ground floor.

The talks on Tuesday will take place in Aula 5 and the talks on  Wednesday will take place in Aula 1 which are located on the ground floor.

The opening session will be live-streamed at the following link: https://polito-it.zoom.us/j/83554741656?pwd=rn94xJbTMSYJubED95OpN8UjcpykIz.1

You can download the titles and abstracts of the talks here

List of talks

1. Marco Andreatta: Variations on Fano's Last Fano - slides

In 1949 Fano published his last paper on $3$-folds with canonical sectional curves (Rendiconti Accademia dei Lincei). There he constructed and described a $3$-fold of the type $X^{22}_3 \subset \mathbb{P}^{13}$ with canonical curve section, which we like to call Fano's last Fano. Together, first with Roberto Pignatelli and thereafter with Ciro Ciliberto, we explored Fano's ingenious construction extending it in various directions in modern language.  

2. Marian Aprodu: Resonance and vector bundles

Resonance varieties are algebro-geometric objects that emerged from the geometric group theory. They are naturally associated to vector subspaces in second exterior powers and carry natural scheme structures that can be non-reduced. In algebraic geometry, they made an unexpected appearance in connection with syzygies of canonical curves. In this talk, based on several works with G. Farkas, Y. Kim, C. Raicu, A. Suciu, C. Spiridon and J. Weyman I report on some recent results concerning the geometry of resonance schemes in the vector bundle setup.

3. Enrique Arrondo: On the notion of regularity of sheaves - slides

We will discuss the notion of regularity of sheaves on projective varieties. The starting point will the work of Costa and Mirò-Roig using n-blocks. We will go beyond their definition, giving an equivalent definition and discussing when it is possible to relax their hypotheses. We will use also the idea of n-blocks to produce other interesting results, like a generalization of ACM bundles and Horrocks Theorem. This is work in progress with Simone Marchesi.

4. Alessandra Bernardi: Generalized additive decompositions and the regularity of schemes - slides

In this talk, I will explore how generalized additive decompositions give rise to specific algebraic schemes and analyze their geometric and algebraic properties. In particular, I will focus on the regularity of these schemes, investigating how their structure is influenced by the underlying decomposition.

5. Maria Chiara Brambilla: Minimal Terracini loci in projective spaces - slides

The notion of Terracini loci has recently been introduced for projective spaces and then extended to other projective varieties. Such objects give information for interpolation problems over double points in special position and are related to the study of special loci contained in secant varieties to projective varieties. I will describe the main known results on this topic, focusing on the case of minimal Terracini loci in projective spaces. This is joint work with Edoardo Ballico.

6. Jarosław Buczyński: Fujita vanishing, sufficiently ample line bundles, and cactus varieties

For a fixed projective scheme X, we say that a property P(L) (where L is a line bundle on X) is satisfied by sufficiently ample line bundles if there exists a line bundle M on X such that for any ample L the property P(L+M) holds. I will discuss which properties of line bundles are satisfied by the sufficiently ample line bundles - for example, can you figure out before the talk, whether a sufficiently ample line bundle must be very ample? The grandfather of such properties and a basic ingredient used to study this concept is Fujita's vanishing theorem, which is an analogue of Serre's vanishing for sufficiently ample line bundles. At the end of the talk I will define cactus varieties (an analogue of secant varieties) and sketch a proof that cactus varieties to sufficiently ample embeddings of X are (set-theoretically) defined by minors of matrices with linear entries. The topic is closely related to conjectures of Eisenbud-Koh-Stillman (for curves) and Sidman-Smith (for any varieties). Based on a joint work with Weronika Buczyńska and Łucja Farnik, https://arxiv.org/abs/2412.00709.

7. Cinzia Casagrande: Classifying Fano 4-folds with large Picard number

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. We will discuss the following theorem: if rho(X)>11, then X is a product of del Pezzo surfaces. This implies, in particular, that the maximal Picard number of a Fano 4-fold is 18. After an introduction and a discussion of examples, we will explain some of the ideas and techniques involved in the proof.

8. Fabrizio Catanese: Finite Gorenstein  coverings in the work of  Casnati and Ekedahl, and recent applications to the moduli spaces of surfaces with p_g=q=2 - slides

In the talk, after describing the results of  the seminal papers by Casnati and Ekedahl, I will try first to describe Gianfranco’s results for covering of higher degree (such as 6), and some old research plans with him that have remained dormient for more than a decade. Then I shall deal with some small generalizations which have been given in a recent work with Massimiliano Alessandro, and some which are object of work in progress with Matteo Penegini, concerning applications  to the moduli spaces of surfaces of general type with  p_g=q=2.

9. Paola Frediani: Asymptotic directions in the moduli space of curves

I will report on a joint work with E. Colombo and G.P. Pirola, where we study asymptotic directions in the tangent bundle of the moduli space of curves of genus g, namely those tangent directions that are annihilated by the second fundamental form of the Torelli map. I will give examples of asymptotic directions for any g  at least 4. I will  show that if the rank d of a tangent direction at [C] (with respect to the infinitesimal deformation map) is less than the Clifford index of the curve C, then the tangent direction  is not asymptotic.  Finally I will determine all asymptotic directions of rank 1 and give an almost complete description of asymptotic directions of rank 2. 

10. Giorgio Ottaviani: The Hessian map - slides

The Hessian map is the rational map associating to each homogeneous polynomial in r+1 variables its Hessian, which is the determinant of its Hessian matrix. The Hessian map is equivariant for the action of SL(r+1). We prove that the Hessian map is birational on its image for r=1, 2 with the exception of a few well known cases. The proof requires the harmonic decomposition of a polynomial,obtained by the action by SO(r+1) and the classification of the integer points of two elliptic curves performed by Jerson Caro and Juanita Duque-Rosero . The result has been obtained jointly with Ciro Ciliberto.

11. Giuseppe Pareschi: Two characterizations of decomposable abelian varieties with a one-dimensional factor

I will show two characterizations of (principally) polarized abelian varieties of the form A=ExB, with E an elliptic curve. The interest of such characterizations is that they both look analogous to known results of curve theory, characterizing hyperelliptic curves. The first one is by means of the non-surjectivity of a certain multiplication map of global sections. The second one is via the non-surjectivity of a certain second gaussian-Wahl map, and can be seen as an effective version of a theorem of Nakamaye on abelian varieties of minimal Seshadri constant. Joint work with Nelson Alvarado.