Vincenzo Antonelli

Ulrich bundles on Hirzebruch surfaces

Abstract. Ulrich bundles on a projective variety are vector bundles that admit a completely linear resolution as sheaves on the projective space. They carry many interesting properties and they are the simplest one from the cohomological point of view.

In this talk we characterize Ulrich bundles of any rank on polarized rational ruled surfaces over \(\mathbb{P}^1\). We show that every Ulrich bundle admits a resolution in terms of line bundles. Conversely, given an injective map between suitable totally decomposed vector bundles, we show that its cokernel is Ulrich if it satisfies a vanishing in cohomology.

Finally, we discuss some particular cases and we construct examples of indecomposable Ulrich bundles.

Giulio Caviglia

Maximal syzygies in Hilbert schemes of complete intersections

Abstract. Let \(d_1,\ \ldots,\ d_c\) be positive integers and consider the monomial complete intersection \(Y = \mathrm{proj}\big( \Bbbk[x_1, \ldots, x_{n+1}]/(x_1^{d_1}, \ldots, x_{c}^{d_{c}})\big)\subseteq \mathbb{P}^{n}\). For each Hilbert polynomial \(p(\zeta)\) we construct a distinguished point in the Hilbert scheme \(\mathrm{Hilb}^{p(\zeta)}(Y)\), which we call the expansive point. This point achieves the largest possible syzygies among all subschemes \(Z \in \mathrm{Hilb}^{p(\zeta)}(Y)\). Assuming the validity of the Lex-plus-powers conjecture, the expansive point provides uniform sharp upper bounds for the syzygies of subschemes \(Z \in \mathrm{Hilb}^{p(\zeta)}(X)\) for all complete intersections \(X = X(d_1, \ldots, d_{c}) \subseteq \mathbb{P}^{n}\). In some cases, the expansive point achieves extremal Betti numbers for the infinite free resolutions associated to a subscheme in \(\mathrm{Hilb}^{p(\zeta)}(Y)\). Our approach is new even in the special case \(Y = \mathbb{P}^{n}\), where it provides new results and simpler proofs of known theorems. This is a joint work with Alessio Sammartano.

Luca Chiantini

On the geometry of symmetric tensors

Abstract. Using tools of algebraic geometry for the analysis of finite subsets of projective spaces, it is possible to carry on a detailed analysis of symmetric tensors (i.e. polynomial forms). Many properties of tensors which are relevant for computational purposes follow from an application of advanced techniques for the study of Hilbert functions. I will survey on recent achievements in the theory.

Alessandro De Stefani

Globalizing invariants in positive characteristic

Abstract. The Fröbenius endomorphism plays a crucial role in the study of the singularities of rings in positive characteristic. Starting from this map, one can define several invariants that allow to detect how severe the singularities of a local ring are. With the goal of extending this study to objects such as coordinate rings of algebraic varieties, we develop a framework that allows to extend these notions to the non-local setting. In particular, in this talk we will focus on how to “globalize” Hilbert-Kunz multiplicities and, time permitting, Fröbenius Betti numbers. This is joint work with Thomas Polstra and Yongwei Yao.

Andreas Hochenegger

Fröbenius neighborhoods of cubic fourfolds

Abstract. Given a functor between triangulated categories which admits both adjoints, I present a way how to measure the difference between those adjoints. This leads in the case of fully faithful functors to the notion of a Fröbenius neighborhood.

But this talk will not be solely categorical, I will also show how to apply this approach to the Kuznetsov component of a cubic fourfold. This is joint work in progress with Ciaran Meachan.

Samuele Mongodi

A geometric intuition for the zeroes of slice-regular functions

of a quaternionic variable

Abstract. The theory of slice-regular functions of a quaternionic variable is an attempt to generalize holomorphic functions to the quaternionic setting; the definition was given by Gentili and Struppa in 2006 and a number of equivalent formulations have been found since then. In particular, a slice-regular function is associated to a “usual” holomorphic map, with values in \(\mathbb{C}^4\); this approach, however, has not been fully exploited yet because of the difficulty of understanding the behaviour of the values (for example, the zeroes) of the slice-regular function in terms of the associated holomorphic map.

In this talk, I will present the basics of the theory of slice-regular functions and show how to study its zeroes in terms of the associated holomorphic map.

Time permitting, I will also discuss about the possible generalizations to Clifford algebras (or real alternative algebras), where a notion of slice-regularity is also possible.

Paolo Saracco

An Hopf algebroid approach to jets spaces

and Lie algebroid integration

Abstract. Commutative Hopf algebroids are the algebraic counterpart of groupoids in almost the same way in which commutative Hopf algebras are the counterpart of groups. As such, they may reveal to be a powerful tool in dealing both with algebraic and geometric questions. The aim of this talk is that of introducing very briefly some distinguished types of Hopf algebroids (namely commutative, cocommutative and complete ones) and to present two possible applications of them.

Firstly, let \(A\) be a commutative algebra over a field \(k\) and \(L\) be a Lie-Rinehart algebra over \(A\) whose underlying \(A\)-module \(L\) is finitely generated and projective. For example, take \(L\) to be the \(A\)-module of global sections of a Lie algebroid \(L\) over a smooth connected real manifold \(M\), with \(A = \mathcal{C}_{\mathbb{R}}^{\infty}(M)\). Then we may associate to it two complete commutative Hopf algebroids (formal groupoids). The first one is the completion \((A,\widehat{\mathcal{V}_A(L)^{\circ}})\) of the finite dual commutative Hopf algebroid of its universal enveloping algebra \(\mathcal{V}_A(L)\). The second one will be the “dual” Hopf algebroid \((A,\mathcal{V}_A(L)^\ast)\) appeared in the work of M. Kapranov on the spaces of paths, where \(\mathcal{V}_A(L)^\ast\) is the convolution algebra of \(\mathcal{V}_A(L)\). We will report on the existence of a canonical \((A\otimes A)\)-algebra map \(\zeta : \mathcal{V}_A(L)^\circ \to \mathcal{V}_A(L)^\ast\) which factors through a continuous morphism \(\widehat{\zeta} : \widehat{\mathcal{V}_A(L)^\circ}\to \mathcal{V}_A(L)^\ast\) of complete Hopf algebroids and we will also discuss conditions under which the map \(\widehat{\zeta}\) is an homeomorphism.

Secondly, we will see that the algebra of infinite jets \(\mathcal{J}(A) = \varprojlim_{n} (A \otimes A) / \ker (m)^n\) naturally admits a complete Hopf algebroid structure and we will find a relationship between it and the completion \(\widehat{\text{Diff}(A)}\) of the finite dual commutative Hopf algebroid of the algebra \(\text{Diff}(A)\) of linear differential operators on \(A\) of any order, which we hope may extend the classical duality \(\text{Diff}_k(A) \cong {}^\ast\mathcal{J}_k(A)\) between differential operators of order \(k\) and \(k\)-jets.

Based on a joint project with L. El Kaoutit. ArXiv: 1705.06698, 1705.03433.