Joan Carles Naranjo

Universitat de Barcelona

Some new enumerative properties of cubic threefolds via Prym theory
In this talk we will report about some new enumerative results on a general smooth cubic threefold V, obtained jointly with Martí Lahoz and Andrés Rojas. We consider the Fano surface F(V) of the lines contained in V and the curve Γ={l∈F(V)∣∃ a 2-plane π and a line r∈F(V) with V⋅π=l+2r} parametrizing the lines l∈F(V) whose conic bundle structure has a singular discriminant curve. We prove that Γ is numerically equivalent to 8KF(X) and has exactly 1485 ordinary nodes as singularities. This amounts to say that there are exactly 1485 lines l⊂V for which there exist 2-planes π1,π2⊂P4 and lines r1,r2⊂V satisfying V⋅πi=l+2ri (i=1,2). The first part of the talk will be a survey about classical topics as the non-rationality of V, the Clemens-Griffiths approach via the intermediate Jacobian JV, and the representation of JV as Prym varieties of coverings of plane quintics via conic bundle structures. Then we will introduce the moduli spaces of coverings, their Picard groups and the properties of the Prym maps defined on them. Finally, we will give a rough idea of the proof of the enumerative results.

Giorgio Ottaviani

Università degli Studi di Firenze

The geometry of tensor spaces Slides
Tensors are the multidimensional generalization of matrices and appear everywhere in mathematical models and engineering applications. Matrix rank can be generalized to tensor rank. This new concept is surprisingly quite different and more difficult, but it acquires a uniqueness feature in decompositions that makes it very important in applications. It opens a new perspective about classical results on secant varieties and on the decomposition of a polynomial in sum of powers. We will introduce some of the new features of the tensor world and review some recent results and open problems.

Patricio Almirón Cuadros

Universidad Complutense de Madrid

On the difference between Milnor number and Tjurina number of isolated singularities Slides
In this talk we will talk about the upper bounds for the difference between Milnor number and Tjurina number of isolated singularities. Our main motivation for this topic is the following question posed by Dimca and Greuel in 2017: is the quotient of Milnor and Tjurina numbers less than 4/3 for any plane curve singularity? We will present a complete solution to the Dimca and Greuel's question together with its natural extension to singularities in higher dimensions. This generalization will link this topic with an old standing conjecture about the geometric genus posed by Durfee in 1978.

Guillem Blanco

KU Leuven

Yano's conjecture Slides
In 1982, T. Yano proposed a conjecture about the generic b-exponents of an irreducible plane curve singularity. Given any holomorphic function f:(C2,0)⟶(C,0) defining an irreducible plane curve, the conjecture gives an explicit description for the generic b-exponents of the singularity in terms of the resolution of f. In this talk, we will present a proof of Yano's conjecture.

Yairon Cid-Ruiz

Ghent University

An invitation to the fiber-full scheme
We introduce the fiber-full scheme which can be seen as the parameter space that generalizes the Hilbert and Quot schemes by controlling the entire cohomological data. In other words, the fiber-full scheme controls the dimension of all cohomologies of all possible twistings, instead of just the Hilbert polynomial. We also present some applications that derive from the existence of the fiber-full scheme. This talk is based on joint work with Ritvik Ramkumar.

Liena Colarte Gómez

Universitat de Barcelona

Gröbner’s problem and the geometry of GT-varieties Slides
In this talk, we present new contributions to the longstanding problem, posed by Gröbner, of determining when a monomial projection of the Veronese variety is an aCM variety. We focus on the aCM property and the geometry of a family of monomial projections of the Veronese variety directly related to the invariants of finite abelian groups and the Lefschetz properties of artinian ideals.

Jesús Martín Ovejero

Universidad de Salamanca

The moduli space of principal bundles with formal trivializations Slides
In mathematics, some of the main problems that arise in almost every field of the discipline are those concerning the classification of objects up to an equivalence relation. From a geometric point of view, a moduli problem is just a classification problem such that to give a solution of the problem (a moduli space) is to give a geometric space such that each point of this space corresponds with an equivalence class of the objects we are trying to classify. In this sense, our geometric spaces will be schemes defined over the spectrum of an algebraically closed field of characteristic zero. In this talk, we explain the moduli problem of principal G-bundles equipped with a formal trivialization over an algebraic projective curve C and we construct the moduli space BunG,C∞. We relate our construction with the stack of principal G-bundles , obtaining a uniformization theorem that allow us to express the stack of principal bundle as the quotient stack of BunG,C∞ by a canonical action of the positive loop group of G. Finally, we introduce the notion of pgg-algebra and we explain how to obtain a canonical embedding of the moduli space BunG,C∞ into a projective bundle.

Carlos Jesús Moreno Ávila

Universitat Jaume I

Non-positive at infinity valuations of Hirzebruch surfaces and Newton-Okounkov bodies
In this talk we consider plane divisorial valuations of Hirzebruch surfaces and introduce the concept of non-positivity at infinity. We prove that the surfaces given by valuations of the last types have nice global and local geometric properties. Moreover, non-positive at infinity divisorial valuations are those divisorial valuations of Hirzebruch surfaces providing rational surfaces with minimal generated cone of curves. Finally, we compute the Seshadri-type constants for pairs formed by a big divisor and a divisorial valuation of a Hirzebruch surface and obtain the vertices of the Newton-Okounkov bodies of pairs as above under the non-positivity at infinity property.

Cedric Oms

Ecole Normale Supérieure de Lyon

First steps in b-contact topology
b-contact structures are a generalization of contact structures where the non-integrability condition fails along a hypersurface. In this talk I will talk about first steps towards a classification of b-contact structures in dimension 3 in terms of homotopy classes of plane fields that are tangent to a given hypersurface, thus generalizing a celebrated result of Eliashberg. This is based on work in progress with Robert Cardona.

María de la Paz Tirado Hernández

Universidad de Sevilla

Leaps of the integrability (in the sense of Hasse-Schmidt) Slides
In the first part I will recall the main definitions and properties of Hasse-Schmidt derivations, focusing on the case in which the base ring has positive characteristic. In the second part, I will give some results related to leaps of the chain of modules of m-integrable derivations (in the sense of Hasse-Schmidt).

Andrés Rojas

Universitat de Barcelona - Universität Bonn

Chern degree functions Slides
Given a smooth polarized surface, we will introduce Chern degree functions associated to any object of its derived category. These functions encode the behaviour of the object along the boundary of a certain region of Bridgeland stability conditions. We will discuss their extension to continuous real functions and the meaning of their differentiability at certain points. These functions turn out to be especially interesting for abelian surfaces, as they recover the cohomological rank functions defined by Jiang and Pareschi. In the final part we will see geometric applications of this equivalence.
Universitat de Barcelona RGAS