Programme and abstracts

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Homemade recipe for apolar natural schemes
Alessandra Bernardi (Università degli studi di Trento)

Take a homogeneous polynomial.
Select a good enough variable that you will dehomogenize the polynomial with respect to.
Whisk the polynomial in the derivative space. Keep whisking until the polynomial is reduced to nothing.
The detail that changes everything: As you whisk, remember to set aside the derivatives that contributed to the cancellation of the polynomial. These derivatives will be those that you will use to build your scheme.
Now your scheme is ready to be served.
To enhance the flavor and add some zest to your scheme, serve it with a glass of refreshing Split. The unique taste of Split will complement the mathematical elegance of your scheme.
For an extra touch, you can also add a drizzle of cactus sauce to your scheme. This sauce will provide a tang(ens)y and sharp flavor, further enhancing the overall experience.

Problems and related results in Algebraic Vision and  Multiview Geometry
Marina Bertolini /  Cristina Turrini  (Università degli studi di Milano)

An  interesting class of  projective algebraic varieties arises naturally in the setting of Computer Vision, in connection with reconstruction problems from a set of linear projections P^k - - > P^{h_j}, j=1,…n.  The key tool to study these varieties is a family of tensors, introduced by R.Hartley and F. Shaffalitzky, called Grassmann tensors. Determinantal varieties which are critical for the reconstruction can be studied leveraging these tensors: we will describe their ideal and provide some classification results.

Grassmann tensors are objects of interest on their own: in the case n=3, we determine their rank, multilinear rank, and core; in the case n=2, we give a birational description of the variety of these tensors.

These results are based on joint works with other co-authors: GianMario Besana, Gilberto Bini, and Roberto Notari.


Duality of cones for Mori Dream Space
Chiara Brambilla (Università Politecnica delle Marche) 

Given a Mori Dream Space, I will describe the cones of divisors which are ample in codimension k, proving that they are rational polyhedral.

Then I will explain the duality between such cones and the cones of k-moving curves. In general a Weak Duality property holds, already proved by Payne and Choi. I will show that a Strong Duality property is satisfied by the family of Mori dream spaces given by Blow-ups of projective spaces at general points.

This is a joint work with Dumitrescu, Postinghel and Santana-Sanchez.


Ulrich bundles on Fano threefolds
Ciro Ciliberto (Università degli studi Tor Vergata, Roma)

In this talk, after having recalled some general facts about Ulrich bundles, I will focus on the study of Ulrich bundles on smooth Fano threefolds. I will illustrate some results on the existence of such stable bundles on these threefolds also discussing dimension and smoothness of the corresponding moduli spaces. This is joint work in collaboration with F. Flamini and A. L. Knutsen.


ACM bundles on a Grassmannian
Laura Costa (Universitat de Barcelona)

In the talk I’ll characterize homogeneous arithmetically Cohen–Macaulay (ACM, for short) bundles on Grassmannians
Gr(k, n)  by means of the description of some matrices that I’ll associate to them. In addition,  I’ll construct irreducible families of indecomposable ACM bundles on  Gr(k,n).


Foliations on homogeneous spaces
Daniele Faenzi (Université de Bourgogne)

A codimension-1 foliation on a projective manifold X can be though of as a corank-1 (saturated) subsheaf F of the tangent bundle TX which is stable under the Lie bracket. After fixing the determinant of F, the set of such foliations is a locally closed subset of the space of 1-forms on X with values in some fixed line bundle L. We investigate the set of these foliations when X is a projective homogeneous space of Picard rank 1, for the simplest possible choice of L. A special focus will be on Grassmannians, most notably Grassmannians of lines, where our foliations are particularly easy to describe. In collaboration with V. Benedetti and A. Muniz.


Ulrich Bundles on  scrolls over F_e
Lucia Fania (Università degli studi dell'Aquila)

I will report on a joint work with Flaminio Flamini, in which we investigate the existence of Ulrich vector bundles on some 3-fold scrolls X_e over Hirzebruch surfaces F_e, with e ≥ 0, which arise as tautological embeddings of  projectivization of very-ample vector bundles  on F_e that are uniform in the sense of Brosius and Aprodu--Brinzanescu.

 We  describe components of moduli spaces of rank r ≥ 1 vector bundles  which are Ulrich with respect to the tautological polarization on X_e and whose general point is a slope-stable, indecomposable vector bundle. We moreover determine the dimension of such components, proving also that they are generically smooth. 

Triple solids and scrolls
Antonio Lanteri (Università degli studi di Milano) 

(joint project with Carla Novelli)
Let Y be a smooth projective variety of dimension n ≥ 2 endowed with a finite morphism \phi Y --> P^n of degree 3, and 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.


Lefschetz properties and Perazzo hypersurfaces
Emilia Mezzetti (Università degli studi di Trieste)

Artinian Gorenstein algebras (AG algebras for short) can be viewed as algebraic analogues of the cohomology rings of smooth projective varieties. The Strong and Weak Lefschetz properties for graded AG algebras take origin from the hard Lefschetz theorem. The properties of an AG quotient A _F of a polynomial ring are related to its Macaulay dual generator F, and in particular A_F fails the Strong Lefschetz property if and only if the hessian of F of order t vanishes for some 1≤ t ≤ d/2, where d=deg F and the usual hessian is obtained for t=1. Perazzo polynomials are a large class of polynomials with vanishing hessian so their algebras A_F always fail the Strong Lefschetz property. I will report on some recent results concerning the question if the Weak Lefschetz property holds  for these algebras, focusing on the case of at most five variables. We will see that the answer only depends on the Hilbert function of A_F: it results to be always unimodal and we characterize those Hilbert functions for which the Weak Lefschetz property holds.

This is joint work with N. Abdallah, N. Altafi, P. De Poi, L. Fiorindo, T. Iarrobino, P. Macias Marques, R.M. Miró-Roig, L. Nicklasson.

Gröbner's problem
Rosa María Miró-Roig (Universitat de Barcelona), EMS Distinguished Speaker

In my talk, I will address Gröbner’s problem. It is a longstanding open problem in commutative algebra and algebraic geometry, posed by W. Gr¨obner in 1969 and it aims to determine whether a (monomial) projection of a Veronese variety is an arithmetically Cohen-Macaulay variety. I will summarize what is known about this problem and explain some recent contributions.


Vector bundles without intermediate cohomology, but with Enrique insight
Giorgio Ottaviani (Università degli studi di Firenze)

Horrocks proved in 1964 that vector bundles on P^n without intermediate cohomology split as direct sum of line bundles. This result has been the starting point of a great research activity on aCM varieties, showing interesting connections with derived categories and other areas. Enrique Arrondo and his school have given several contributions to this topic. We try to follow some paths into this fascinating story, which has classical roots.


Some special families of general curves
Edoardo Sernesi (Università degli studi Roma Tre)

In  the 1990's  A. Treibich and J.L. Verdier  discovered a class of curves of genus g, for any g ≥3, which enjoy some remarkable properties. In particular Treibich proved that they are Brill-Noether general.  I will  survey the main properties of such curves and outline Treibich's proof of their Brill-Noether generality.


Computation of the p-adic Abel-Jacobi map of the product
of a Hilbert modular surface and a modular curve
Ignacio Sols (Universidad Complutense de Madrid)

(joint project with Ivan Blanco-Chacón)
We compute the value of the p-adic Abel-Jacobi map of the product of the moduli of abelian surfaces with multiplication by the integers of a real quadratic field, with suitable polarization and level, and the modular curve naturally embedded in it, this evaluated at a null-homological modification of the embedding of the curve in this product, a value which will be related in later work to the critical value of an L-function.


A Saito criterion for several polynomials
Jean Vallès (Université de Pau et des Pays de l'Adour)

In this work in progress with D. Faenzi and M. Jardim, we propose to generalize Saito’s criterion, that characterizes the freeness of a projective hypersurface, to any projective subvariety.


Projecting a Grassmannian of lines from its Calabi-Yau sections.
Alessandro Verra (Università degli studi Roma Tre)

The first part of the talk revisits, historically as well, some line geometry of Pfaffian cubic fourfolds in a complex 5-dimensional projective space and the related family of K3 surfaces of genus 8. Then, moving from the seminal paper of Beauville-Donagi, the case is considered of hypersurfaces defined by a (2r+2) x (2r+2) Pfaffian of linear forms, in a projective space of dimension 2r+1. Some natural generalizations are presented,  involving the Grassmannian of lines G(1, 2r+1) and its Calabi-Yau linear sections of codimension 2r+2. Joint work with Michele Bolognesi.