Abstracts

Title: Local cohomology through derivators theory

Abstract: Since its appearance in Grothendieck-Hartshorne notes, local cohomology has become an essential tool for working in algebraic geometry. The local cohomology functors can be defined as a derived torsion functor with support in closed sets and it is important to have general methods to compute them.

For a commutative noetherian ring R, Neeman established a correspondence between subsets of Spec(R) and localizing subcategories of the derived category D(R). In particular, the smashing subcategories are in correspondence with specialization-closed subsets. Cohomology with support in such subsets is a natural ganeralization of the classical case and expresses the also classical idea of substituting a closed subset by a system of supports.

We will give a novel description of the local cohomology functors in this case, using derivators to take advantage of the nice properties of Koszul complexes.

Title: Slope inequalities for irregular fibrations.

Abstract: Given an irregular fibration from a smooth complex projective variety X onto a curve B, I will show how to obtain lower bounds for the slope of a line bundle L using the so called Clifford-Severi inequalities and how to classify the limit cases. Moreover, we will see that Clifford-Severi inequalities and Slope inequalities for a given class o fibrations are equivalent, via Pardini's trick and a continuous version of Xiao's method.

Celia del Buey de Andrés, Universidad Autónoma de Madrid

Title: Differentiably simple rings and ring extensions defined by p-basis.

Abstract: In this talk, we will review the concept of differentiably simple ring, in particular, Harper’s Theorem on the characterization of Noetherian differentiably simple rings in positive characteristic from a new perspective. Then we will study flat families of differentiably simple rings, or equivalently, finite flat extensions of rings which locally admit p-basis. These extensions, which were studied by Yuan, are called Galois extensions of exponent one. For such an extension A ⊂ C, we will introduce an A-scheme, called the Yuan scheme, which parametrizes subextensions A ⊂ B ⊂ C such that B ⊂ C is Galois of a fixed rank. So, roughly, the Yuan scheme will be thought of as a kind of Grassmannian of Galois subextensions. Finally, we will study some geometric properties of the Yuan scheme (as smoothness and irreducibility) and compute the dimension of the fibers.

This is joint work with Diego Sulca (FAMAF, Universidad Nacional de Córdoba, Argentina) and Orlando Villamayor (Universidad Autónoma de Madrid).

Antonio Campillo, Universidad de Valladolid

Title: Poincaré series and geometric combinatorics

Abstract: By revisiting several results due to Ann Lemahieu, we present some bridges between Poincaré series, used in Singularities, and monomial generating functions, used in Geometric Combinatorics. Those bridges allow to construct Poincaré series describing geometric properties of certain combinatorial objects.

Title: Quasi-projective varieties with orbifold fundamental groups.

Abstract: In this talk, we’ll study geometric conditions for a quasi-projective variety to have a free product of cyclic groups as its fundamental group. As a consequence, we obtain that every curve in CP^2 whose fundamental group of its complement is isomorphic to the free product of Z_p and Z_q (with p, q>1 coprime) is given by a polynomial of the form (f_p)^q+(f_q)^p, where f_p and f_q are homogeneous polynomials of degrees p and q with no common factors. Our methods also allow us to produce curves such that the fundamental groups of their complements are free products of cyclic groups, generalizing Oka’s classic examples. Joint work with José Ignacio Cogolludo Agustín.

Juan Elias, University of Barcelona

Title: Sumsets and and Veronese varieties.

Abstract: We present a link between the additive combinatorics of subsets of Z^n and the geometry of Veronese varieties.

In the case n=1, we associate to a finite set of non-negative integers A a monomial projective curve C_A such that the Hilbert function of C_A and the cardinalities of the sumsets sA agree. Moreover, the singularities of C_A determine the asymptotic behavior of the function s-->|sA|. We mainly focus on the case n=1, if we have time, we will comment some results for a general integer n.

Evelia García Barroso, Universidad de La Laguna

Title: On the equisingularity class of the general higher order polars of plane branches.

Abstract: We describe the factorization of the higher order polars of a generic branch in its equisingularity class. We generalize the results of Casas-Alvero "On the singularities of polar curves (1983)" and Hefez-Hernandes-Hernández "On the factorization of the polar of a plane branch (2018).

This is a joint work with Janusz Gwoździewicz and Mateusz Masternak.

Title: Archimedean zeta functions and $b$-functions for meromorphic functions.

Abstract: In this talk we give the definition and properties of the Archimedean zeta function associated to a quotient of analytic functions.

We also present a family of Bernstein-Sato polynomials (or b-functions) for those quotients, the functional equations that they satisfy as well as some generalizations of classical results by Kashiwara and Lichtin. Finally we give some generalizations of multiplier ideals and jumping numbers in this setting. This is a joint work with Josep Àlvarez Montaner, Manuel González Villa and Luis Núñez Betancourt.

Marina Logares Jiménez, Universidad Complutense de Madrid

Title: The symplectic geometry of Higgs bundles with poles.

Abstract: A Higgs bundle consist of an holomorphic bundle over a Riemann surface together with a Higgs field. These objects are in the core of gauge theories, being solutions to a dimensional reduction of Yang-Mills equations, and the geometry of their moduli spaces has been studied extensively for the past decades. Here we will focus on the case of Higgs bundles on non-compact Riemann surfaces, which introduces poles on the Higgs field and even a more richer geometry. In this talk we shall analyse their Poisson structure as well as the integrable systems encoded. We shall not assume any previous knowledge on Higgs bundles, so we shall also provide an overview on the subject. This is based on previous and ongoing work with Biswas, Martens, Peón-Nieto and Szabo.

Jesus Martinez Garcia, University of Essex

Title: The Calabi Problem: Kaehler-Einstein metrics, K-stability and the description of compact moduli of Fano varieties in low dimensions.

Abstract: According to the Minimal Model Programme, Fano varieties are one of the building blocks of all other projective varieties. Unlike other building blocks, a renowned result by C. Birkar guarantees that if we 'bound' their singularities, Fano varieties belong to a finite number of deformation families. Thus, their classification is theoretically within reach. Moduli spaces are the standard proto-tool to classify varieties, however the moduli of Fanos does not behave well due to their (possibly) large automorphism groups, even for smooth Fano varieties.

The solution by Chen-Donaldson-Sun to the Yau-Tian-Donaldson conjecture, and subsequent generalisations due to C. Li and Liu-Xu-Zhuang established that the existence of a Kahler-Einstein metric on a Fano variety is equivalent to the algebro-geometric notion of K-polystability. Moreover, the work of Liu-Xu-Zhuang confirmed that K-polystable Fano varieties form a projective moduli space (known as K-moduli), thus giving the best candidate so far for moduli for Fano varieties. All the above said, determining which specific Fano varieties are K-polystable and how its moduli space looks like is a formidable problem, in which birational geometry plays a significant role. In this talk I will survey some of the recent developments in this area and focus on the study and detection of K-polystability in dimension 3, including the classification of which of the 105 deformation of smooth Fano threefolds have a general K-polystable element, recently established with my collaborators.

Title: Curves with prescribed multiplicities and Bounded Negativity.

Abstract: We will show a lower bound on the degree of curves of the projective plane passing, with prescribed multiplicities, through the centers of a divisorial valuation. Also, we will provide some results related to the Bounded Negativity Conjecture concerning rational surfaces having the projective plane as a relatively minimal model. This talk will be based on joint work with C. Galindo, C. J. Moreno Ávila and E. Pérez Callejo.

Guillermo Peñafort Sanchis, Universidad de Valencia

Title: TBA

Abstract: TBA

Elvira Pérez Callejo, Universidad Jaume I

Title: Algebraic integrability of foliations on Hirzebruch surfaces: An algorithm for the existence and computation of rational first integrals of fixed genus.

Abstract: To decide whether a plane foliation has a rational first integral is an open problem. We provide an algorithm that, under certain conditions, decides about algebraic integrability of fixed genus which, in the affirmative case, computes a rational first integral. This algorithm uses the extension of the foliation to a Hirzebruch surface and the reduction of its dicritical singularities.

Francisco Plaza Martín, Universidad de Salamanca

Title: Kummer's theory for the geometric adeles of a curve.

Abstract: This talk deals with abelian extensions of the function field of an algebraic curve over an algebraically closed field of arbitrary characteristic. The study of abelian extensions is a classical topic that has been approached from many different perspectives (Galois theory for finite extensions of a field, Class field theory for local and global fields). Specifically, a Kummer theory for the ring of adeles of the function field of an algebraic curve will be presented and, as an application, compared with the corresponding Kummer theory of the defining function field.

Eamon Quinlan Gallego, University of Utah

Title: The Bernstein-Sato polynomial in positive characteristic.

Abstract: The Bernstein-Sato polynomial of a holomorphic function is an invariant that originated in complex analysis, and with now strong applications to birational geometry and singularity theory over the complex numbers. For example, it detects the log-canonical threshold as well as the eigenvalues of the monodromy action on the cohomology of the Milnor fibre. In this talk I will survey some results on a characteristic-p analogue of this invariant.

Andrés Rojas, Humboldt University, Berlin

Title: Cohomological rank functions on abelian surfaces via Bridgeland stability.

Abstract: In the context of abelian varieties, Z. Jiang and G. Pareschi have introduced interesting invariants, cohomological rank functions, associated to Q-twisted (complexes of) coherent sheaves.

We will show that, in the case of abelian surfaces, Bridgeland stability provides an alternative description of these functions. This helps to understand their general structure, and allows to compute geometrically meaningful examples. As a main application, we will give new results on the syzygies of abelian surfaces.

This is a joint work with Martí Lahoz.

Antonio Rojas León, Universidad de Sevilla

Title: Estimates for generalized Jacobi sums

Abstract: Jacobi sums are classically defined in terms of multiplicative characters of finite fields and are closely related to Gauss sums, and their absolute values are well known. We will define a generalization of these sums and illustrate how to use algebro-geometric techniques (such as l-adic cohomology and perverse sheaves) to obtain optimal estimates for them.

Edison Sampaio, Universidade Federal do Ceará

Title: Lipschitz regular complex analytic sets are affine linear subspaces

Abstract: In this talk, I will present some Bernstein-type results. I will prove that a pure-dimensional complex analytic set which is bi-Lipschitz homeomorphic to an Euclidean space (outside of compact sets) must be an affine linear subspace. A non-parameteric version of this result will be also presented.

Javier Sánchez González, Universidad de Salamanca

Title: Lax colimits of posets with structure sheaves: applications to descent.

Abstract: Schematic finite spaces constitute a certain subcategory of finite ringed posets that models qc-qs schemes and other more general spaces. In this talk, motivated by the study of the descent-like properties of certain invariants defined on these spaces, we will introduce, for any 1-category or strict 2-category C, the category of finite posets with a structure C-valued functor: the category of C-data.

We will analyze the 2-categorical properties of C-data and explicitly compute certain lax colimits in it that we shall call «cylinders» and that do not require us to perform any operations on either C or the category of posets. Among other applications, the specialization of these cylinders to the schematic case will be «weakly equivalent» to the ordinary colimit, which will allow us to prove results such as a general Seifert-Van Kampen Theorem for an «schematic étale fundamental group» with relative ease, extending the homonym result for schemes to a weaker topology.

Luis José Santana Sánchez, Universidad de La Laguna

Title: The geometry of ideals of fat points.

Abstract: Let R be the polynomial ring in n+1 variables and let I be an homogeneous ideal of a collection of s fat points of R, it is an open question to compute the Hilbert function at degree d of such an ideal. Geometrically, this corresponds to computing the dimension of L, the linear system of hypersurfaces of degree d in the complex projective space of dimension n, passing through the s points with prescribed multiplicities. The problem of computing this dimension is known in the literature as the polynomial interpolation problem, since it relates to a question of Hermite interpolation. In general, the Hilbert function at degree d of a given ideal of fat points I is expected to agree with its Hilbert polynomial evaluated in d. When these values do not match we say that the corresponding linear system L is special.

The goal of this talk is to understand the geometry behind the speciality of linear systems. We will give an overview on the subject highlighting the most famous results known to date. Then, we will present our result which gives full answer to the interpolation problem when the points sit on a rational normal curve of degree n, which we denote by C. In order to do so, we will translate the question to the study of global sections in X, the blow-up of the projective space at the given s points . It is known due to Castravet and Tevelev that, for points in C, the variety X is a Mori dream space. We will see that the speciality of linear systems with support on C can be completely understood by the presence of certain varieties in their base locus. It turns out that these varieties are part of the walls in the Mori chamber decomposition of X.

Ilya Smirnov, BCAM

Title: Signatures for singularities in positive characteristic

Abstract: The Frobenius endomorphism can be used in many ways to study singularities in positive characteristic, for example, it is used to define various classes of singularities and various measures of singularities. In this talk, I will present two fundamental classes, F-regular and F-rational singularities, and discuss the measures of singularities associated to them.

The presented new results are from a joint work with Kevin Tucker.