Schedule 2021

Thirty years of mirror symmetry

John Alexander Cruz Morales (Universidad Nacional de Colombia)

Abstract: In this talk we will give an overview of the different flavors of mirror symmetry since the publication of the seminal paper by Candelas-de la Ossa-Green-Parkes in 1991. We will describe the main ideas developed during the last 30 years and will try to give some perspectives for the future.

Slides & Video

Linked projective spaces

Renan da Silva Santos (IMPA)

Abstract: A limit linear series over a nodal curve with n components naturally gives rise to the structure called linked net of vector spaces. It is a quiver representation over a special type of quivers, the Z^n-quivers, with properties that capture the geometry from the limit linear series. To any exact linked net of vector spaces g with finite support we can associate a projective scheme LP(g), called linked projective space, which parametrizes sub-representations of dimension 1. In this talk we briefly review the theory of limit linear series and introduce the formal concept of linked nets of vector spaces. Our main result is the following:
If g is an exact linked net of vector spaces with finite support over a Z^2-quiver, then the scheme LP(g) is pure dimensional and all the components are rational. If time permits, we also give an explicit description of the components using an equivalence relation in the vertices of the base quiver and a structural theorem that classifies all linked nets of vector spaces of dimension 1.

Slides & Video

A modular approach to the class number problem

Felipe de León Saenz Angel (IMPA)

Abstract: In this talk I will present a modular approach to the class number problem due to J.-P. Serre. In short, I will explain the relationship between imaginary quadratic number fields having class number equal to one and a certain class of points in a suitable class of modular curves, and how this point of view allows us to solve a particular case of class number problem.

Slides & Video

No seminar

Hodge Locus

Jorge Armando Duque Franco (Universidad de Antioquia)

Abstract: The Hodge locus was introduced by Grothendieck to study the Hodge Conjecture in families. Even in the 2-dimensional case where the Hodge conjecture is known to hold, its components are far from being understood. In this talk, I will give an imprecise historical tour of what is known about the Hodge Locus. And if time permits I will comment a little about some strange Hodge cycles that appear when trying to understand a certain Hodge Locus (work in progress with R. Villaflor).

Slides & Video

Finite subgroups of PGL(2, K)

Abraham Rojas Vega (ICMC - USP)

Abstract: In this work we study the finite subgroups of PGL(2, K), where K is an algebraically closed field of positive characteristic. These groups act in K(x) 3-sharply transitively. We use the Theory of Algebraic Function Fields in one variable, specially the concept of ramified extension. We show an example of the computations involved. We also give a "generic example" of each subgroup, i.e., a subfield of K(x) that is the fixed field by the current subgroup. This work is based on [1].

REFERENCES

[1] Valentini R., Manohar R. A Hauptsatz of L.E Dickson and Artin-Schreier extensions, Journal für die Reine und Angewandte Mathematik, 1979.

[2] Stichtenoth H. Algebraic Functions Fields and Codes, Springer, 2009.

Slides & Video

No seminar

Rational elliptic surfaces over number fields

Felipe Zingali Meira (UFRJ/University of Groningen)

Abstract: We say that a surface is elliptic if it is endowed with a surjective morphism to a curve such that almost every fibre is an elliptic curve. When an elliptic surface over an algebraically closed field is rational, it is isomorphic to the blow-up of the projective plane in the base points of a pencil of cubics. When we look at these surfaces over a non-algebraically closed field, the situation branches, and in this seminar we will explore some of the different possibilities.

Slides & Video

Jouanolou's theorem

Gabriel Fazoli Domingos (IMPA)

Abstract: When we are going to solve an ODE, there is nothing better than finding an algebraic solution. In the context of foliations in surfaces, this translates to finding invariant algebraic curves. In this sense Jouanolou's theorem appears, which states that most foliations of degree greater than or equal to 2 in the projective plane do not have any invariant algebraic curve. In this seminar, we will introduce some basic concepts of foliations and demonstrate Jouanolou's theorem.

Slides & Video

The conjectures of Mahler and Viterbo

Cláudia Rodrigues da Silveira (IMPA)

Abstract: The Mahler conjecture is a famous problem in Convex Geometry, at the moment we have positive solutions in dimensions 2 and 3, proven in 1939 and 2020, respectively. On the other hand, there is a totally open problem in Symplectic Geometry known as the Viterbo Conjecture. We will learn how these problems are related through the use of billiards games.

Slides & Video

The monodromy problem for hyperelliptic curves

Daniel Felipe López Garcia (Tsinghua University)

Abstract: We study the Dynkin diagram associated with the monodromy of direct sums of polynomials. The monodromy problem asks under which conditions on a polynomial, the monodromy of a vanishing cycle generates the whole homology of a regular fiber. We consider the case y⁴+g(x), which is a generalization of the results of Christopher and Mardešić about the monodromy problem for hyperelliptic curves. Moreover, We solve the monodromy problem for direct sums of fourth-degree polynomials.

Slides

Gauss-Manin connection in Disguise: K3 surfaces

Walter Andrés Páez Gaviria (IMPA)

Abstract: In this talk I will recall some of the basic ideas of Movasati's program, called Gauss-Manin connection in Disguise. In particular, I will explain how we can obtain the algebra of Siegel modular forms of genus two by applying this program to the case of K3 surfaces. As a by-product, we obtain a generalization of Siegel modular forms that we call Siegel quasi-modular forms.

A theorem by Nagata

Eduardo Alves da Silva (IMPA)

Abstract: This will be an expository talk about a theorem by Nagata which asserts that the surface obtained by blowing up the complex projective plane at 9 or more very general points contains infinitely many exceptional curves, the so-called (-1)-curves. I will explore some facts about the geometry of blowups of the complex projective plane and I will give two slightly different ways to show this result as well the reason behind this infiniteness phenomenon which does not occur when we blow up less than 9 very general points.

Slides

Standard conjectures and Hodge theory for matroids

Manoel Zanoelo Jarra (IMPA)

Abstract: Matroids are combinatorial objects that generalize the concept of independence. To each graph we can associate a matroid, and each matroid has its characteristic polynomial, the analogue of the chromatic polynomial in graph theory. In this talk we will see how the standard Hodge-type conjecture inspired the answer to an old conjecture about the distribution of characteristic polynomials coefficients.

Slides

Non-conservative function fields

Cesar Augusto Hilario Poma (IMPA)

Abstract: A function field in one variable can be seen as the field of rational functions of an algebraic curve. These function fields are important because they make it possible to study algebraic curves from an intrinsic point of view, putting emphasis on the curve itself rather than on its ambient space. Furthermore, several results from the theory of algebraic curves, such as the Riemann-Roch theorem, can be deduced from their analogues in the setting of function fields.
In this talk I will give an overview of the theory of function fields in one variable. The emphasis will be placed on the so-called non-conservative function fields. I will explain in particular –if time permits– how they can be used to study fibrations by singular curves in positive characteristic.

Slides

Tutte polynomial via K-Theory

Manoel Zanoelo Jarra (IMPA)

Abstract: Matroids are combinatorial objects that express the idea of linear independence. Each subspace of a vector space has an associated matroid, and each matroid has its Tutte polynomial, which encodes properties related to constraint and contraction operations. In this presentation I will show how to obtain a geometric description of this polynomial using Grassmannian K-Theory.

Slides

Classification of semisimple Lie algebras

Diego Salazar Gutierrez (IMPA)

Abstract: Lie algebras are algebraic objects that appear in the study of Lie groups in differential geometry, in quantum mechanics in physics, and other areas. We give an overview of the classification theory of semisimple Lie algebras in the most transparent case: when the base field is algebraically closed and with characteristic 0. In this case, the families of Lie algebras are given by Dynkin diagrams. We will also see some connections of this theory with classical algebraic geometry.

Slides

Mordell-Weil rank jumps on elliptical surfaces

Renato Dias (UFRJ)

Abstract: A surface X is elliptic if there is a curve C and a morphism X -> C whose fibers are almost all elliptic curves. Essentially, it is a surface formed by a family of elliptic curves, which well suggests its relevance to arithmetic geometry. A natural question is about the behavior of the Mordell-Weil rank of elliptic curves along the surface. In this presentation we will talk about ways to approach the problem and recent results on rank jumps.

Slides

Geometry of Higher Rank Valuations

Hernan Iriarte (École Polytechnique)

Abstract: It is well known since before Zariski that the set of (equivalent classes) of valuations on the function field of an algebraic curve is in correspondence with the points of the curve. In higher dimensional varieties this picture gets more complicated, nonetheless it is still important in many situations. In this talk we will give tools to understand geometrically the space of full rank valuations on function fields of algebraic varieties. The approach will be through the study of valuations of a simple kind called monomial valuations. In particular, we will get that the space of higher rank valuations can be obtained as a limit of certain tangent bundles of polyhedrons.
The talk will require only a basic course in algebraic geometry.

Slides

No seminar

Logarithmic components of spaces of singular projective foliations

Javier Gargiulo (UFRJ)

Abstract: First, we will describe moduli algebraic spaces parametrizing singular foliations. In particular, we will focus on the case of singular projective foliations of codimension one. Then, the "stability problem" of these foliations via algebraic methods will be discussed. As a consequence, we are going to prove the stability of codimension one logarithmic foliations using Zariski tangent space calculations. This corresponds to a joint work with Prof. Cukierman, F. and Prof. Massri, C. and determines an algebraic proof of a result due to Prof. Calvo Andrade, O. Finally, we will briefly describe how to extend part of these results to higher codimensional foliations and some open problems.

Slides

Cremona group and G-birational superrigidity of Del Pezzo surfaces

Lucas das Dores (IMPA)

Abstract: The classification of finite subgroups up to conjugation of the plane Cremona group was completed by Iskovskikh and Manin. One of the main ideas for this classification is to associate a finite subgroup G of the Cremona Group to an action of a group of automorphisms on a rational surface, resulting in a G-surface. This classification can be refined with the concepts of G-birationally rigid and superrigid surfaces, not yet completely classified. In this lecture, we present these concepts and a joint result with M. Mauri providing the complete classification of G-birationally superrigid Del Pezzo surfaces of degree less than or equal to 3.

Slides

Birational geometry of blow-ups of projective spaces

Inder Kaur (PUC Rio)

Abstract: The Gale correspondence provides a duality between sets of n general points in projective spaces P^s and P^r when n equals r + s + 2. By a result of Mukai, the blow-up of P^4 at 8 points say X, can be realized as a moduli space of torsion-free rank 2 semi-stable sheaves (with certain fixed Chern class datum) on the blow-up of P^2 in 8 Gale dual points. In a recent work, Casagrande, Codogni and Fanelli use this to describe the Mori chamber decomposition of the effective cone of divisors of X. It was shown by Castravet and Tevelev that the blow-up of P^r at n points for the case when r ≥ 5 and n ≥ r + 4 is no longer a Mori dream space. In joint work with Carolina Araujo, Ana-Maria Castravet and Diletta Martinelli we show that even in this case it is possible to give a Mori chamber type decomposition for a part of the effective cone.

Slides

An example of an accumulation point of volumes due to Blache

Diana Torres (PUC Chile)

Abstract: Our ground field is the complex numbers. Let W be a stable surface, and let K its canonical class. A fundamental result is the Descending Chain Condition (DCC for short) for {K^2}, which is due to Alexeev. A full description of {K^2} and Acc({K^2}) is still an open problem. In this talk, we recall some definitions and results in such a context. Also, we construct a sequence of stable surfaces {W_k} such that 1 belongs to Acc({K_{W_k}^2}). That construction is due to Blache, and it starts by blowing up a configuration of lines in the projective plane.

Slides

Beyond quasi-modular forms from a geometric point of view

Roberto Villaflor (PUC Chile)

Abstract: In the first part of this talk I will give a quick overview of classical modular forms interpreted in terms of moduli spaces of elliptic curves. In the second part I will explain Movasati's geometric approach to quasi-modular forms, and how this program works for much more general moduli spaces (e.g. of genus two curves, K3 surfaces, Abelian varieties, Calabi-Yau varieties) providing us with a wealthy source of interesting functions going beyond classical quasi-modular forms.

Slides

Z_p-towers of curves

Roberto Alvarenga (ICMC-USP)

Abstract: Let p be a prime number and X be a nonsingular projective curve defined over a finite field of q=p^r elements. While Weil's theorem, or Riemann's hypothesis for curves over finite fields, states that the roots of the Weil polynomial have a complex norm of \sqrt{q}, the p-adic norms of these roots remain somewhat mysterious in general. In this seminar we will study the behavior of such p-adic norms when X traverses a Z_p-tower.

Slides

No seminar

Hilbert property: from the inverse Galois problem to the topology of manifoldspology

Ana Quedo (IMPA)

Abstract: A smooth algebraic manifold V defined over a field k satisfies the Hilbert property when the set of its k-rational points is not lean in the Serre sense. Lean sets are, by definition, given by finite unions of Zariski closed proper and overlay images of degree greater than or equal to 2. In this lecture, we will talk a little about the historical motivation of this property and its connections with algebraic topology and with the inverse Galois problem.

Slides

Classification of antiassociative algebras

Crislaine Kuster (IMPA)

Abstract: The classification of n-dimensional algebras defined by polynomial identities is a classic problem in the theory of non-associative algebras. Therefore, it is common to focus on low dimensions and, in this case, there are two directions to follow: algebraic classification and geometric classification.
On the other hand, anti-associative algebras, i.e., algebras that satisfy the identity x(yz) + (xy)z = 0, have been gaining visibility in several scientific works, due to their relationship with other types of algebras, such as Mock-Lie algebras and anticommutative algebras. Therefore, in this presentation we will focus on the algebraic and geometric classification of anti-associative algebras of dimension up to 5.

Slides

No seminar

The Cremona group

Daniela Paiva (IMPA)

Abstract: The n-Cremona group Bir(P^n) is the group of birational automorphisms of P^n. The study of the structure of Bir(P^n) is a classical problem that goes back to the 19th century in the celebrated Noether-Castelnuovo Theorem: Every birational automorphism of P^2 is a composition of projective linear transformations and the standard quadratic transformation. In higher dimensions things become more complicated. Hilda Hudson and Ivan Pan proved that any set of group generators of the n-Cremona group, n ≥ 3, contains uncountably many transformations of unbounded degree. More recently, J. Blanc, S. Lamy and S. Zimmermann (2019) showed that Bir(P^n), n ≥ 3, is not generated by Aut(P^n), the Jonquière maps and any subset with cardinality smaller than that of complex numbers.
In this talk, I will discuss this last result and a powerful tool, taken from the MMP, used for its proof: The Sarkisov Program, which provides a decomposition of any birational automorphism of P^n into elementary links, called Sarkisov links.

Slides

Linked nets and linked projective spaces

Renan da Silva Santos (IMPA)

Abstract: A linked net of vector spaces is a representation to special type of quivers, called the Z_n-quivers. When n = 1, those linked nets are precisely the linked chains of vector spaces introduced by Rocha. Linked chains are a natural generalization of limit linear series over compact type curves with two components. Linked nets are an attempt to transport this generalization to compact type curves with more components. We focus on curves with three components. We show a theorem that completely characterize all linked nets of dimension one.
To each linked net g we associate the linked projective scheme LP(g). In a sense, it is a generalization of the scheme P(g) studied by Esteves and Osserman. When n = 1, Rocha proved that the scheme LP(g) is Cohen–Macaulay, reduced, of pure dimension r = rank of g. It is also a deformation of the diagonal inside a product of projective spaces.
Using the characterization of linked nets of dimension one, we prove that LP(g) is also reduced of pure dimension r, for the case n = 2. If time permits, we also show that for r = 1 and r = 2, that scheme also have the same Hilbert polynomial of the diagonal.

Slides

Foliations and reduction modulo p

Wodson Mendson (IMPA)

Abstract: Given a foliation F in a complex projective variety X we can apply the reduction modulo p process to obtain a foliation F_p in X_p, a variety over a field of characteristic p > 0. Information about the foliation modulo p can imply very interesting information about the original foliation. In fact, this is the source of many open problems. The great example consists of the Grothendieck-Katz conjecture which basically consists of a local-global principle for the existence of algebraic solutions in ordinary differential equations. Our objective will be to understand how to use reduction modulo p to study foliation problems in complex varieties focusing on a result of local nature on the existence of holomorphic first integrals and another of global nature on non-algebraicity of foliations in the complex plane P^2.

Slides

No seminar

Linear limit series of curves with three components

Eduardo Vital (IMPA)

Abstract: The concept of limit linear series was defined by David Eisenbud and Joe Harris in 1986 (Limit linear series: Basic theory) in order to study linear series via deformations from smooth curves to singular curves. In a natural process of attempts at generalizations, some new concepts and difficulties emerged. We will briefly talk about the existence of a simple basis and about the linked projective space of an exact representation associated with a curve with three components of the non-compact type.

Slides

Linear limit series of curves with three components

João Paulo Lindquist Figueredo (IMPA)

Abstract: Let y' = p(x,y) be an ordinary differential equation in the complex plane. We know that a unique solution passes through each point in the plane. An interesting question is to determine when the solutions are algebraic curves, that is, given as zeros of polynomials. One way to do this is to compact the complex plane into a projective variety X and consider the extension of this ODE in X. This extension defines a foliation in X, which is nothing more than a global object in X that is locally given by a system of differential equations. The solutions of this system are called leaves of the foliation. In this talk, I will address the problem of determining when leaves are algebraic varieties. Conjecturally, every foliation without singularities in a rationally connected projective variety has algebraic leaves. I will show a partial result about this conjecture for foliations on 3-dimensional varieties.

Slides

Cremona transformations as hyperbolic isometries

Luíze D'Urso (PUC Rio/IMPA)

Abstract: Cremona transformations are simply birational maps of P^n into P^n. We are interested in the case n = 2, and we want to understand well the Cr(P^2) group of these transformations, known as the Cremona Group of P^2.
By the Nöether-Castelnuovo Theorem (late 19th century), the Cremona Group of P^2 is generated by the automorphisms of P^2 and by the Cremona Transformation Φ given by (x∶ y∶ z) ↦ (yz∶ xz∶ xy). But although we understand the group Aut(P^2) ≅ PGL(3) well, when we add the element Φ, the generated group becomes much more complicated.
Only in 2013, Cantat and Lammy proved that Cr(P^2) is not simple in the case of an algebraically closed field. In 2016, Anne Lonjou proved the same for any body. Both tests are based on an action by isometrics of the Cremona Group of P^2 in an infinite-dimensional hyperbolic space. Our objective will be to understand this action.

Slides

Decomposition and inertia groups and birational geometry of Calabi-Yau pairs

Eduardo Alves da Silva (IMPA)

Abstract: Decomposition and inertia groups are special subgroups of the plane Cremona group which preserve a certain irreducible curve as a set and pointwise, respectively. Castelnuovo, and more recently, Jérémy B., Ivan P. and Thierry V. have proved a bunch of interesting results and provided some descriptions of them. In the particular case where the fixed curve is a nonsingular cubic, we have the notion of Calabi-Yau pair which allows us to use new tools to deal with the study of these groups. I will show how to use this different approach to recover a result by Ivan P. and deduce some very nice properties of birational plane maps that preserve such nonsingular cubic.

Slides