researchseminars.org

Gaia Comaschi (University of Campinas)
2020-04-15 18:30 - 19:30 UTC | https://stable.researchseminars.org/talk/BRAG/1/
Title: GIT stability of linear systems of skew-symmetric forms (https://stable.researchseminars.org/talk/BRAG/1/)
by Gaia Comaschi (University of Campinas) as part of Brazilian algebraic geometry seminar

Abstract
Given a 6 dimensional vector space $W$, we consider $\mathbb{P}(\mathbb{C}^{n+1}\otimes \bigwedge ^2 W^*)$, the projective space parameterizing n-dimensional linear systems of skew-symmetric forms on $W$. Since the group $SL(W)$ acts on $\mathbb{P}(\mathbb{C}^{n+1}\otimes \bigwedge ^2 W^*)$, Geometric Invariant Theory (GIT) provides a notion of (semi)stability. In this talk I will introduce a criterion to detect the (semi)stability of linear systems of skew-symmetric forms and I will then present how this criterion allows to obtain a complete classification of all stable linear systems having generic rank equal to 4.

Access the Zoom link
https://zoom.us/j/92887438541?pwd=[REDACTED]

Giosuè Muratore (Federal University of Minas Gerais (UFMG))
2020-04-29 18:30 - 19:30 | https://stable.researchseminars.org/talk/BRAG/2/
Title: A Recursive Formula for Osculating Curves (https://stable.researchseminars.org/talk/BRAG/2/)
by Giosuè Muratore (Federal University of Minas Gerais (UFMG)) as part of Brazilian algebraic geometry seminar

Abstract
Let $X$ be a smooth complex projective variety. Using a construction
devised to Gathmann, we present a recursive formula for some of the
Gromov-Witten invariants of $X$. We prove that, when $X$ is homogeneous, this
formula gives the number of osculating rational curves at a general point of a general hypersurface of $X$. This generalizes the classical well known pairs of in inflection (asymptotic) lines for surfaces in $\mathbb{P}^3$ of Salmon, as well as Darboux's 27 osculating conics.

Link for the talk will be provided a few days in advance.

Lucas das Dores (IMPA)
2020-04-22 18:30 - 19:30 | https://stable.researchseminars.org/talk/BRAG/3/
Title: Schemes of rational curves on Del Pezzo surfaces (https://stable.researchseminars.org/talk/BRAG/3/)
by Lucas das Dores (IMPA) as part of Brazilian algebraic geometry seminar

Abstract
Schemes parametrizing rational curves on a projective variety have a natural partition in terms of the degrees of the rational curves. In this talk, we present a natural refinement of this partition on schemes parametrizing rational curves on Del Pezzo surfaces. The classical description of Del Pezzo surfaces as blow-ups of the projective plane at points in general position yields that these refined partitions reflect the multiplicity of the rational curves at each of the blown-up points. Moreover, we compute the dimension of components of these parameter spaces containing (points corresponding to) resolutions of plane curves which are singular at the blown-up points.

Ugo Bruzzo (SISSA/UFPB)
2020-05-13 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/5/
Title: About the McKay correspondence in 3 dimensions (https://stable.researchseminars.org/talk/BRAG/5/)
by Ugo Bruzzo (SISSA/UFPB) as part of Brazilian algebraic geometry seminar

Abstract
Abstract: I will review work on the McKay correspondence in 3 dimensions and its differential-geometric version, relying on papers by Ito-Reid, Sardo Infirri, Craw-Ishi and others, and on some original results obtained with a group of collaborators. This is about the resolution of singularities of the type C^3/G, where G is a finite subgroup of GL(3,C) or SL(3,C). I will also discuss the chamber structure for the stability parameter of the GIT quotient. I will illustrate the general theory by means of a nontrivial but manageable example (C^3/Z_4 with Z_4 acting as a subrgoup of SL(3,C)). I will also hint at some physical motivations.

Link for the Zoom meeting: https://us02web.zoom.us/j/84979113106?pwd=[REDACTED]

Charles Almeida (Federal University of Minas Gerais (UFMG))
2020-05-20 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/6/
Title: The geography of moduli spaces of torsion free sheaves (https://stable.researchseminars.org/talk/BRAG/6/)
by Charles Almeida (Federal University of Minas Gerais (UFMG)) as part of Brazilian algebraic geometry seminar

Abstract
In this talk I will describe new irreducible components of the moduli space of rank 2 semistable torsion free sheaves on the 3-dimensional projective space, whose generic point corresponds to non-locally free sheaves. As an application, I will compute the number of irreducible components of the moduli space of torsion free sheaves, with first, second and third Chern classes equal to -1, 2 and 2 respectively. Additionaly, I will give an idea of how to prove that this moduli space is connected. This is a joint work with Marcos Jardim and Alexander S. Tikhomirov.

Tiago Fonseca (University of Oxford)
2020-06-03 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/7/
Title: From transcendental numbers to higher Ramanujan foliations (https://stable.researchseminars.org/talk/BRAG/7/)
by Tiago Fonseca (University of Oxford) as part of Brazilian algebraic geometry seminar

Abstract
I will explain how a problem in the theory of transcendental numbers leads to the construction of certain principal bundles over moduli stacks of abelian varieties. Such bundles carry a natural horizontal foliation whose corresponding differential equations generalize Ramanujan's classical relations between Eisenstein series. I will then discuss a result on the Zariski-density of the analytic leaves of this foliation.

Google meet link https://meet.google.com/wwv-sajj-fsn

Jorge Vitório Pereira (IMPA)
2020-05-27 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/8/
Title: Miyaoka's algebraicity criterion and variations (https://stable.researchseminars.org/talk/BRAG/8/)
by Jorge Vitório Pereira (IMPA) as part of Brazilian algebraic geometry seminar

Abstract
I will review some old and new results/arguments on the
algebraicity of leaves of foliations with "positive" tangent
sheaf.

Link for google meet will be posted a few days in advance.

Herivelto Borges, Saeed Tafazolian, Luciane Quoos Conte and Cícero Carvalho (Federal University of Minas Gerais (UFMG))
2020-06-10 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/9/
Title: In honor of Fernando Torres (https://stable.researchseminars.org/talk/BRAG/9/)
by Herivelto Borges, Saeed Tafazolian, Luciane Quoos Conte and Cícero Carvalho (Federal University of Minas Gerais (UFMG)) as part of Brazilian algebraic geometry seminar

Abstract
Our esteemed colleague Fernando Torres passed away on May 28th 2020, aged 58. He completed his PhD at IMPA in 1993 under the supervision of Arnaldo Garcia, and was a professor at the University of Campinas since 1998. He was widely known for his many contributions to the theory of algebraic curves over finite fields and its applications.

This memorial session will have short presentations by Herivelto Borges, Saeed Tafazolian, Luciane Quoos Conte and Cícero Carvalho and completed with testimonies by Arnaldo Garcia, and Torres's students, collaborators and friends.

Link for google meet: meet.google.com/bve-gahp-ocn

Cecília Salgado (Federal University of Rio de Janeiro (UFRJ))
2020-06-24 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/10/
Title: Mordell-Weil rank jumps and the Hilbert property (https://stable.researchseminars.org/talk/BRAG/10/)
by Cecília Salgado (Federal University of Rio de Janeiro (UFRJ)) as part of Brazilian algebraic geometry seminar

Abstract
Let X be an elliptic surface with a section defined over a number field. Specialization theorems by Néron and Silverman imply that the rank of the Mordell-Weil group of special fibers is at least equal to the MW rank of the generic fiber. We say that the rank jumps when the former is strictly larger than the latter. In this talk, I will discuss rank jumps for elliptic surfaces fibred over the projective line. If the surface admits a conic bundle we show that the subset of the line for which the rank jumps is not thin in the sense of Serre. This is joint work with Dan Loughran (Bath).

Link for the google meet will be posted here a few days before the talk.

Genival da Silva Junior (Imperial College)
2020-06-17 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/11/
Title: Surfaces with Exceptional monodromy (https://stable.researchseminars.org/talk/BRAG/11/)
by Genival da Silva Junior (Imperial College) as part of Brazilian algebraic geometry seminar

Abstract
There have been several constructions of family of varieties with exceptional monodromy group. In most cases, these constructions give Hodge structures with high weight(Hodge numbers spread out). N. Katz was the first to obtain Hodge structures with low weight(Hodge numbers equal to (2,3,2)) and geometric monodromy group G2. I this talk I will give an alternate description of Katz's result using Hodge theory.

André Contiero (Federal University of Minas Gerais (UFMG))
2020-07-01 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/12/
Title: Curves and Weierstrass points (https://stable.researchseminars.org/talk/BRAG/12/)
by André Contiero (Federal University of Minas Gerais (UFMG)) as part of Brazilian algebraic geometry seminar

Abstract
In this talk we will present some working in progress on the moduli space of pointed curves with prescribed Weierstrass semigroup at the marked point. We will present a tight lower bound for its dimension when the semigroup is non-negatively graded, and we will also give sufficient condition to its rationality when the semigroup is symmetric.

Link for google meet will be posted here a few days in advance.

Amar Henni (Federal University of Santa Catarina (UFSC))
2020-07-22 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/13/
Title: On the fixed locus of framed instanton sheaves on $\mathbb{P}^3$ (https://stable.researchseminars.org/talk/BRAG/13/)
by Amar Henni (Federal University of Santa Catarina (UFSC)) as part of Brazilian algebraic geometry seminar

Abstract
Let T be the three dimensional torus acting on $\mathbb{P}^3$ and MT(c) be the fixed locus of the corresponding action on the moduli space of rank 2 framed instanton sheaves on $\mathbb{P}^3$. We show that MT(c) consist only of non locally-free instanton sheaves whose double dual is the trivial bundle. Moreover, we relate these instantons to multiple structures and give a classification of their support. This allows to compute a lower bound on the number of components of MT(c).

Link for the google meet will be posted here a few days before the talk.

Rick Richster (Federal University of Itajubá (UNIFEI))
2020-07-08 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/14/
Title: Secant defectiveness of toric varieties (https://stable.researchseminars.org/talk/BRAG/14/)
by Rick Richster (Federal University of Itajubá (UNIFEI)) as part of Brazilian algebraic geometry seminar

Abstract
The $h$-secant variety $Sec_{h}(X)$ of a non-degenerate $n$-dimensional variety $X\subset\mathbb{P}^N$ is the Zariski closure of the union of all linear spaces spanned by collections of $h$ points of $X$.
The expected dimension of $Sec_{h}(X)$ is
$Expdim(Sec_{h}(X)):= \min\{nh+h-1,N\}$.
The actual dimension of $Sec_{h}(X)$ may be smaller than the expected one.

Let $N$ be a rank $n$ free abelian group and $M$ its dual. Let $P\subseteq M_{\mathbb Q}$ be a full dimensional lattice polytope and $X_P$ the corresponding toric variety.

In this talk we discuss a new technique to give bounds on the Secant Defectivity of $X_P$ using information from the polytope $P$. It is a joint work just submitted with Antonio Laface and Alex Massarenti.

The link for the google meet will be posted here a few days in advance.

Maral Mostafazadehfard (Federal University of Rio de Janeiro (UFRJ))
2020-07-15 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/15/
Title: Divisor class group of Hankel determinantal rings (https://stable.researchseminars.org/talk/BRAG/15/)
by Maral Mostafazadehfard (Federal University of Rio de Janeiro (UFRJ)) as part of Brazilian algebraic geometry seminar

Abstract
Hankel determinantal rings arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves. In any characteristic we give an explicit description of divisor class groups of these rings and as a consequence we show that they are $\mathbb{Q}$-Gorenstein rings. It has been shown that each divisor class group element is the class of a maximal Cohen Macaulay module.

Based on a joint work with Aldo Conca, Anurag K. Singh and Matteo Varbaro.

Rodrigo Gondim (UFRPe)
2020-08-12 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/16/
Title: Waring problems and the Lefschetz properties (https://stable.researchseminars.org/talk/BRAG/16/)
by Rodrigo Gondim (UFRPe) as part of Brazilian algebraic geometry seminar

Abstract
We study three variations of the Waring problem for homogeneous polynomials, concerning the Waring rank, the border rank and the cactus rank of a form. We show how the Lefschetz properties of the associated algebra affect them. The main tool is the theory of mixed Hessians and Macaulay-Matlis duality. We construct new families of wild forms, that is, forms whose cactus rank, of schematic nature, is bigger then the border rank, defined geometrically.
(Joint with T. Dias)

Roberto Villaflor (IMPA)
2020-08-05 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/17/
Title: Constructing algebraic cycles on hypersurfaces, an explicit approach to Hodge conjecture (https://stable.researchseminars.org/talk/BRAG/17/)
by Roberto Villaflor (IMPA) as part of Brazilian algebraic geometry seminar

Abstract
Hodge conjecture is one of the major conjectures in algebraic geometry. In all the cases where Hodge conjecture has been verified, no constructive proof has been given up to the date. In other words there is no hint about how to construct an algebraic cycle from a given Hodge cycle. In this talk we will consider this problem in the case of hypersurfaces of the projective space. We will explain how this question becomes more treatable when the Hodge cycle is given in a good enough format in terms of Griffiths basis. Reducing the problem to constructing these nice representatives of Hodge cycles. We will see some examples where this approach works and highlight the difficulties that appear in the general case.

Rodrigo Barbosa (Simons Center for Geometry and Physics)
2020-07-29 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/18/
Title: String Dualities, Higgs Bundles and $G_2$ Geometry (https://stable.researchseminars.org/talk/BRAG/18/)
by Rodrigo Barbosa (Simons Center for Geometry and Physics) as part of Brazilian algebraic geometry seminar

Abstract
Dualities in string/M theory often provide novel perspectives for deformation problems in geometry. In one such instance, involving Large $N$ duality in the B-model, one can construct a family of ALE-fibered Calabi-Yau threefolds over a Riemann surface $S_g$ ($g \geq 2$) whose Jacobian integrable system is isomorphic to the $G$-Hitchin system over $S_g$, where $G$ is the compact real form associated to the ALE type via the McKay correspondence. I will explain a different physical framework, involving M-theory/Type IIA duality, that gives an analogous construction of ALE-fibered $G_2$-manifolds parametrized by spectral covers of certain "smooth Higgs bundles" over a $3$-manifold. I will explain how this theory connects with Donaldson's theory of Kovalev-Lefschetz fibrations and how it presents a window for applying algebro-geometric techniques to moduli problems in $G_2$-geometry. Time permitting, I will comment on a second algebro-geometric model for the moduli space of (complexified) $G_2$-structures derived from SYZ Mirror Symmetry for the Type IIA geometry.

Eduardo Esteves (IMPA)
2020-09-02 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/19/
Title: Degenerations of line bundles along curves (https://stable.researchseminars.org/talk/BRAG/19/)
by Eduardo Esteves (IMPA) as part of Brazilian algebraic geometry seminar

Abstract
A family of line bundles along a family of smooth curves parameterized by the punctured disk can be extended in several ways over the limit stable curve of the family. We show that the collection of all extensions can be naturally parameterized by the torus quotient of the arrangement of toric varieties associated to a certain polytope decomposition of a certain Euclidean space. We characterize all polytope decompositions arising this way in terms of combinatorial data of the stable curve. At the end I will describe how these results may be used to construct new compactifications of Jacobians of stable curves and address the problem raised by Eisenbud and Harris of constructing a useful moduli of limit linear series over the moduli of stable curves. This is an ongoing joint work with Omid Amini (École Polytechnique).

Nuno Cardoso / Aline Zanardini (University of Miami / University of Pennsylvania)
2020-08-26 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/20/
Title: Towards a new technique to compute Orlov spectra / Stability of Halphen pencils of index two (https://stable.researchseminars.org/talk/BRAG/20/)
by Nuno Cardoso / Aline Zanardini (University of Miami / University of Pennsylvania) as part of Brazilian algebraic geometry seminar

Abstract
Towards a new technique to compute Orlov spectra, by Nuno Cardoso

Abstract: A generator of a triangulated category is an object from which we can obtain the whole category through certain operations. Associated to a generator, there is the notion of the generation time, which is the number describing how long the rebuilding process takes. The generation time of the fastest generator is called the Rouquier dimension of the category and it is conjectured that the Rouquier dimension of the derived category of a smooth projective variety of dimension n is exactly n. Orlov suggested that in order to extract additional geometric information from the category, one should study all possible generation times – the Orlov spectrum. Later, Ballard, Favero and Katzarkov developed considerably our understanding of the topic, making connections to rationality, computing the Orlov spectrum in several cases and finding bounds for it. In this talk, we will review part of their results and discuss our work in progress on a new technique to compute the Orlov spectrum, which takes inspiration on Abouzaid's criterion for generating the Fukaya category in terms of open-closed maps.

--xx--xx--

Stability of Halphen pencils of index two, by Aline Zanardini

Abstract: In this talk I will present some results about the stability, in the sense of geometric invariant theory, of Halphen pencils of index two under the action of SL(3). These are pencils of plane curves of degree six having nine (possibly infinitely near) base points of multiplicity two. Inspired by the work of Miranda on pencils of plane cubics, I will explain how to explore the geometry of the associated rational elliptic surfaces. I will also show that the log canonical threshold plays an important role. This work is part of my PhD thesis at the University of Pennsylvania under the supervision of Antonella Grassi.

Alex Abreu (UFF)
2020-08-19 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/21/
Title: A geometric interpretation for characters of Iwahori--Hecke algebras (https://stable.researchseminars.org/talk/BRAG/21/)
by Alex Abreu (UFF) as part of Brazilian algebraic geometry seminar

Abstract
The Iwahori-Hecke algebra is a deformation of the group algebra of the symmetric group. It has a distinguished basis (enumerated by permutations) called the Kazhdan-Lusztig basis. For each permutation we consider certain subvarieties of the complete flag variety that generalize Hessenberg varieties. These varieties carry an action of the symmetric group on its intersection cohomology groups. We prove that the Frobenius character of this action is precisely the Frobenius character of an element of the Kazhdan-Lusztig basis of the Hecke algebra. This is a generalization to non-codominant permutations of Brosnan-Chow's solution to the Sharesian-Wachs conjecture. Some partial results in other Lie types are also achieved.

Link for the google meet will be posted here a few days before the talk.

Alan Muniz (UFF)
2020-09-09 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/22/
Title: On moduli spaces of rank two logarithmic connections over an elliptic curve. (https://stable.researchseminars.org/talk/BRAG/22/)
by Alan Muniz (UFF) as part of Brazilian algebraic geometry seminar

Abstract
We will give an explicit description of the (coarse) moduli space of rank two logarithmic connections with fixed spectral data over an elliptic curve. Precisely, we will show that this moduli space has a covering whose members are easily described.

Joint work with Thiago Fassarella (UFF) and Frank Loray (IRMAR).

Daniele Faenzi (Université de Bourgogne)
2020-09-16 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/23/
Title: Stability of logarithmic tangents (https://stable.researchseminars.org/talk/BRAG/23/)
by Daniele Faenzi (Université de Bourgogne) as part of Brazilian algebraic geometry seminar

Abstract
To a hypersurface $D$ of projective $N$-space $\mathbb{P}^N$ one attaches the \emph{log
tangent sheaf} $T_D$ of vector fields of $\mathbb{P}^N$ tangent to D. For some highly
special hypersurfaces, such as, for instance, hyperplane arrangements
associated to reflection groups and discriminants of binary forms, the
sheaf $T_D$ splits into line bundles - $D$ is then called \emph{free}. On the
other hand, Dolgachev--Kapranov proved that $T_D$ is stable if $D$ is a
generic arrangement of at least $N+2$ hyperplanes; also Dimca proved that
$T_D$ is stable if $D$ has isolated singularities with sufficiently small
Tjurina number and $N=3$.

In this talk we will first show that $T_D$ is stable for a much wider
class of hypersurfaces having low-dimensional singularities. In the
second part of the talk we will prove that $T_D$ is stable if $D$ is the
determinant of $n\times n$ matrices.
If time allows, we will discuss the application from the equisingular
Hilbert scheme containing $D$ to the moduli space of semistable sheaves
containing $T_D$ and show that it is birational in the case of determinants.

This is a report on work in progress with S. Marchesi - soon on the arXiv.

Alessia Mandini (UFF)
2020-09-23 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/24/
Title: Quasi-parabolic Higgs bundles and null hyperpolygon spaces (https://stable.researchseminars.org/talk/BRAG/24/)
by Alessia Mandini (UFF) as part of Brazilian algebraic geometry seminar

Abstract
Hyperpolygons spaces are a family of hyperkähler manifolds, that can be obtained from coadjoint orbits by hyperkähler reduction. Jointly with L. Godinho, we showed that these spaces are isomorphic to certain families of parabolic Higgs bundles, when a suitable condition between the parabolic weights and the spectra of the coadjoint orbits is satisfied.

In analogy to this construction, we introduce two moduli spaces: the moduli spaces of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over $\mathbb{C}\mathbb{P}^1$ on one hand and the null hyperpolygon spaces on the other, and establish an isomorphism between them.

Finally we describe the fixed loci of natural involutions defined on these spaces and relate them to the moduli space of null hyperpolygons in the Minkowski 3-space.

This is based on joint works with Leonor Godinho.

Sally Andria / César Hilário (UFF / IMPA)
2020-09-30 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/25/
Title: An explicit resolution of the Abel map via tropical geometry / Bertini’s theorem in positive characteristic (https://stable.researchseminars.org/talk/BRAG/25/)
by Sally Andria / César Hilário (UFF / IMPA) as part of Brazilian algebraic geometry seminar

Abstract
An explicit resolution of the Abel map via tropical geometry, by Sally Andria (UFF)

In this talk I will talk about the problem studied in my thesis: Is it possible to find an explicit resolution of the Abel map (a rational map) for a nodal curve?
We start with a family of curves, that is a regular smoothing of a nodal curve with smooth components. We take a polarization, an invertible sheaf, and a section through the smooth locus of the family. The Abel map is the rational map taking a tuple of points $(Q_1,\ldots,Q_d)$ on a curve of the family to the associated sheaf in the Esteves compactified Jacobian. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. The solution of the combinatorial problem gives rise to an explicit resolution of the Abel map.

-- xx -- xx --

Bertini’s theorem in positive characteristic, by Cesar Hilario (IMPA)

A classical theorem of Bertini states that in characteristic zero almost all the fibers of a dominant morphism between two smooth algebraic varieties are smooth, that is, there do not exist fibrations by singular varieties with smooth total space. Unfortunately, Bertini’s theorem fails in positive characteristic, as was first observed by Zariski in the 1940s. Investigating such a failure naturally leads to the classification of its exceptions. By a theorem of Tate, a fibration by singular curves of arithmetic genus $g$ in characteristic $p > 0$ may exist only if $p \le 2g + 1$. When $g = 1$ and $g = 2$, these fibrations have been studied by Queen, Borges Neto, Stohr and Simarra Cañate. A birational classification of the case $g = 3$ was started by Stohr ($p = 7, 5$), and then continued by Salomão ($p = 3$). In this talk I shall report on some progress in the case $g = 3$, $p = 2$. In fact, several examples show already that in this setting very interesting geometric phenomena arise.

Damiano Testa (University of Warwick)
2020-10-07 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/26/
Title: Contact in algebraic and tropical geometry (https://stable.researchseminars.org/talk/BRAG/26/)
by Damiano Testa (University of Warwick) as part of Brazilian algebraic geometry seminar

Abstract
In recent years, classical enumerative problems in algebraic geometry have been converted into statements in tropical geometry. This approach has had tremendous success. In view of the current pandemic, we will stay away from these popular results. Rather, we discuss two isolated cases: the 9 inflection points of plane cubics and the 28 bitangent lines of plane quartics. The tropical counts yield 3 and 7, respectively. We will see how to reconcile these results via positive characteristic. These cases naturally generalize to inflection points of plane curves of arbitrary degree and theta-characteristics of curves of general type.

The talk assumes minimal familiarity with basic concepts of algebraic geometry over the complex numbers. Positive characteristic and tropical geometry play important, but non-technical roles. This is joint work with Marco Pacini.

Link for the talk will be posted here a few days before the talk.

Marcos Jardim (UNICAMP)
2020-10-14 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/27/
Title: Classification of codimension one distributions of degree two on the projective space (https://stable.researchseminars.org/talk/BRAG/27/)
by Marcos Jardim (UNICAMP) as part of Brazilian algebraic geometry seminar

Abstract
In this talk I will provide a complete classification of codimension one distributions of degree 2 on the three dimensional projective space, generalizing the classification of codimension one foliations of degree 2 given by Cerveau and Lins Neto. We describe all possible singular schemes and tangent sheaves of such distributions and speculate on the topological and algebraic properties of integrability.

Douglas Guimarães (UNICAMP)
2020-10-28 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/28/
Title: Moduli spaces of quasitrivial rank 2 sheaves (https://stable.researchseminars.org/talk/BRAG/28/)
by Douglas Guimarães (UNICAMP) as part of Brazilian algebraic geometry seminar

Abstract
Douglas Guimarães (UNICAMP)

Title: Moduli spaces of quasitrivial rank 2 sheaves

Abstract: A torsion free sheaf $E$ on $\mathbb{P}^3$ is called quasitrivial if $E^{\vee\vee}=\mathcal{O}_{\mathbb{P}^3}^{\oplus r}$ and $\dim(E^{\vee\vee}/E)=0$. While such sheaves are always $\mu$-semistable, they may not be Gieseker semistable. We study the moduli spaces of $\mu$- and Gieseker semistable quasitrivial sheaves of rank 2 via the quot scheme of points $Quot(\mathcal{O}_{\mathbb{P}^3}^{\oplus 2},n)$, where $n=h^0(E^{\vee\vee}/E)$. We will show the construction of an irreducible component of the Gieseker moduli space which is birrational to the total space of a $\mathbb{P}^{n-1}$-bundle over $S(n-1)\times\mathbb{P}^3$, where $S(n)$ is the smoothable component of the Hilbert scheme of $n$ points in $ \mathbb{P}^3$. Furthermore, this is the only irreducible component when $n\le10$.

Ageu Barbosa (UFPB)
2020-10-21 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/29/
Title: On $(h,s)$-tangential weak defectiveness and identifiability of some projective varieties (https://stable.researchseminars.org/talk/BRAG/29/)
by Ageu Barbosa (UFPB) as part of Brazilian algebraic geometry seminar

Abstract
In this talk, I will present a new technique to study the problem of weak defectiveness using degenerations of tangent linear spaces to osculating linear spaces. I also will present a result on tangential weak defectiveness for varieties admitting a fibration with a linearly embedded $\mathbb{P}^1$ as general fiber and apply it to obtain a sharp asymptotic bound for non secant defectiveness of Segre varieties.

Israel Vainsencher (UFMG)
2020-11-04 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/30/
Title: Enumerative geometry of legendrian foliations (https://stable.researchseminars.org/talk/BRAG/30/)
by Israel Vainsencher (UFMG) as part of Brazilian algebraic geometry seminar

Abstract
Foliations, or more generally, distributions, provide a geometric
viewpoint in the theory of differential equations. Grosso modo,
a foliation of dimension one, is a (polynomial) recipe to draw a line
at each point. We’ll stick to 3-dim projective space. Similarly, a
distribution of codimension one assigns a plane at each point. We
assume the coefficients of the equation of the plane are of degree one.
As syzygies trained minds will recognize, this entails the distribution
is specified by an anti-symmetric 4×4 matrix. Those of maximal
rank correspond to the distributions of contact. A foliation is called
legendrian whenever tangent to some distribution of contact. Our
goal is to describe the calculation of the dimension and degree of
the subvariety of legendrian foliations (and friends). It turns out that
the answer is given by Athus polynomials. This fits into a Schubert
Calculus like programme of exploring the geometry of parameter
spaces of foliations.

Steven Kleiman (MIT)
2021-02-24 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/31/
Title: Geometry of Gorenstein Artinian Algebra (https://stable.researchseminars.org/talk/BRAG/31/)
by Steven Kleiman (MIT) as part of Brazilian algebraic geometry seminar

Abstract
Macaulay Duality, between filtered quotients of a polynomial ring over
a field, annihilated by a power of the variables, and Artinian
submodules of the ring's graded dual, is generalized over any Noetherian
ground ring, and used to provide isomorphisms between the subschemes of
the Hilbert scheme parameterizing various sorts of these quotients, and
the corresponding subschemes of the Quot scheme of the dual. Notably,
on this basis, the scheme of compressed Gorenstein algebras is proved to
be smooth and irreducible of a certain relative dimension. Joint work in progress with Jan Kleppe.

Javier Gargiulo (IMPA)
2020-11-11 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/32/
Title: Rational pullbacks of toric foliations (https://stable.researchseminars.org/talk/BRAG/32/)
by Javier Gargiulo (IMPA) as part of Brazilian algebraic geometry seminar

Abstract
In this talk we will present a short digression about the theory of singular foliations on toric varieties and certain algebraic spaces parametrizing them. In particular, we will construct families of singular foliations on a classical projective space that arise as pull-backs of foliations on a simplicial toric variety X under suitable rational maps. We will focus on the case where X is a complete simplicial toric surface.

The singular set of a foliation is one of the most commonly studied geometric objects in the area. The geometry and topology near a singularity characterize (in some sense) the foliation. Not surprisingly, most of the approaches to obtain stability results for singular foliations involve a detailed study of their singular locus. In this respect, we will attempt to describe certain aspects of the singular and Kupka scheme of foliations on a toric surface and their corresponding pull-backs. We will also characterize their first order unfoldings and deformations in some cases.

Simone Marchesi (Universitat de Barcelona)
2020-11-18 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/33/
Title: Group actions on vector bundles (https://stable.researchseminars.org/talk/BRAG/33/)
by Simone Marchesi (Universitat de Barcelona) as part of Brazilian algebraic geometry seminar

Abstract
The classification of vector bundles which are invariant under the action of a determined group, has been widely studied.

We can consider, for example, the canonical action of the projective linear group $PGL(n+1)$ on $\mathbb{P}^n$ which leads to the definition of homogeneous vector bundle. The choice of specific subgroups has been often determined, in literature, restricting our attention to particular families. Recall indeed that Ancona and Ottaviani proved that the Steiner bundles on $\mathbb{P}^n$ that are invariant under an action of $SL(2,\mathbb{C})$ are the so called Schwarzenberger bundles. Another example, moving into the realm of hyperplane arrangements, in which I have been particularly interested lately, is given by the reflection arrangements. They are defined as hyperplane arrangements that are invariant under the group generated by their reflections, and it is known that their associated sheaf is free (a sum of line bundles) and therefore homogeneous.

In a previous work, studying Nearly-free arrangements, we proved that their configuration of jumping lines is extremely special but remarked that it did not characterize this family of arrangements. It turns out that they are characterized by the invariance under the action of the subgroup $G_p \subset \mathrm{PGL}(3)$ that fixes the point $p$ in the projective plane. Inspired by this result, we classify vector bundles which are invariant under the action of subgroups that fix linear subspaces of the projective plane.

Finally, we will focus on the relations between the geometry of the jumping locus and the invariance under the action of the group. Recall that, historically, such question has been studied in order to relate homogeneous bundles with uniform ones, i.e. bundles for which the splitting type is constant.

This is the result of two collaborations: one with Jean Vallès and one with Rosa Maria Miró-Roig.

Oliver Lorscheid (IMPA)
2020-11-25 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/34/
Title: The moduli space of matroids (https://stable.researchseminars.org/talk/BRAG/34/)
by Oliver Lorscheid (IMPA) as part of Brazilian algebraic geometry seminar

Abstract
Matroids are combinatorial gadgets that reflect properties of linear algebra in situations where this latter theory is not available. This analogy prescribes that the moduli space of matroids should be a Grassmannian over a suitable base object, which cannot be a field or a ring; in consequence usual algebraic geometry does not provide a suitable framework. In joint work with Matt Baker, we have used algebraic geometry over the so-called field with one element to construct such moduli spaces. As an application, we streamline various results of matroid theory and find simplified proofs of classical theorems, such as the fact that a matroid is regular if and only if it is binary and orientable.

We will dedicate the first part of this talk to an exposition of matroids. Then we will briefly outline how to construct the moduli space of matroids. In a last part, we will explain with some care why this theory is useful to simplify classical results in matroid theory.

Carolina Araujo (IMPA)
2020-12-02 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/35/
Title: Birational geometry of Calabi-Yau pairs and 3-dimensional Cremona transformations (https://stable.researchseminars.org/talk/BRAG/35/)
by Carolina Araujo (IMPA) as part of Brazilian algebraic geometry seminar

Abstract
Recently, Oguiso addressed the following question, attributed to Gizatullin: ``Which automorphisms of a smooth quartic K3 surface $D\subset \PP^3$ are induced by Cremona transformations of the ambient space $\mathbb{P}^3$?''

When $D\subset \mathbb{P}^3$ is a smooth quartic surface, $(\mathbb{P}^3,D)$ is an example of a Calabi-Yau pair, that is, a pair $(X,D)$, consisting of a normal projective variety $X$ and an effective Weil divisor $D$ on $X$ such that $K_X+D\sim 0$. Gizatullin's question is about birational properties of the Calabi-Yau pair $(\mathbb{P}^3,D)$. In this talk, I will explain a general framework to study the birational geometry of mildly singular Calabi-Yau pairs. Then I will focus on the case of singular quartic surfaces $D\subset \mathbb{P}^3$. Our results illustrate how the appearance of increasingly worse singularities in $D$ enriches the birational geometry of the pair $(\mathbb{P}^3, D)$, and lead to interesting subgroups of the Cremona group of $\mathbb{P}^3$.

This is joint work with Alessio Corti and Alex Massarenti.

Cristian Martinez (UNICAMP)
2021-03-03 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/36/
Title: Stability under Fourier-Mukai transforms on elliptic surfaces (https://stable.researchseminars.org/talk/BRAG/36/)
by Cristian Martinez (UNICAMP) as part of Brazilian algebraic geometry seminar

Abstract
Let $X$ be a Weierstrass elliptic surface. By moving the polarization towards the fiber direction while keeping the volume of the polarization fixed, we can define a notion of limit Bridgeland stability. In this talk, we will prove that under certain conditions the relative Fourier--Mukai transform of a slope semistable sheaf is a limit semistable object. In the case that the surface has Picard rank two, a detailed study of the potential Bridgeland walls will provide us with extra numerical conditions to guarantee that the Fourier--Mukai transform of a 1-dimensional slope semistable sheaf is Bridgeland semistable.

Zoom Meeting ID: 913 6913 4478

Inder Kaur (PUC-Rio)
2021-03-10 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/37/
Title: Questions on the cohomology ring of moduli spaces of stable locally-free sheaves on curves (https://stable.researchseminars.org/talk/BRAG/37/)
by Inder Kaur (PUC-Rio) as part of Brazilian algebraic geometry seminar

Abstract
Let $M(2,L)$ denote the moduli space of stable vector bundles of rank $2$ and determinant $L$ of odd degree, on a smooth curve of genus $g \geq 2$. Owing to the work of Mumford, Newstead, Kirwan, King and several others, questions such as the generators and relations, higher rank Torelli-type theorems as well as the Hodge conjecture for the cohomology ring of $M(2,L)$ are well understood. In this talk I will survey some of these aspects for the smooth case and discuss analogous results for the case when the underlying curve is irreducible, nodal. This is joint work with Suratno Basu and Ananyo Dan.

Frederico Quallbrunn (Universidad CAECE)
2021-03-17 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/38/
Title: Unfoldings of Lie algebroids (https://stable.researchseminars.org/talk/BRAG/38/)
by Frederico Quallbrunn (Universidad CAECE) as part of Brazilian algebraic geometry seminar

Abstract
We will talk about the notion of unfolding, first on
foliations, and then generalizing this concept to the case of Lie
algebroids. We will show some results and examples appearing in a
joint work with M.Corrêa and A.Molinuevo.

Giosuè Muratore (UFMG)
2021-03-24 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/39/
Title: Enumeration of rational contact curves via torus actions (https://stable.researchseminars.org/talk/BRAG/39/)
by Giosuè Muratore (UFMG) as part of Brazilian algebraic geometry seminar

Abstract
Complex projective spaces of odd dimension have a unique contact structure. So, in these spaces we have contact (Legendrian) rational curves. We are interested in enumeration of such curves. We prove that some Gromov-Witten numbers associated to rational contact curves in projective space with arbitrary incidence conditions are enumerative. Also, we use Bott formula on the Kontsevich space to find the exact value of those numbers.

Danny Taboada / Victor Pretti (UFF / UNICAMP)
2021-03-31 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/40/
Title: The moduli space of quasistable spin curves / Rank 0 Asymptotic Bridgeland stability (https://stable.researchseminars.org/talk/BRAG/40/)
by Danny Taboada / Victor Pretti (UFF / UNICAMP) as part of Brazilian algebraic geometry seminar

Abstract
Young BRAG

Speaker 1: Danny Taboada (UFF)

Title: The moduli space of quasistable spin curves
Abstract: We study a compactification of the moduli space of theta characteristics, giving a modular interpretation of the geometric points and describing the boundary stratification. This space is different from the moduli space of spin curves. The modular description and the boundary stratification of the new compactification are encoded by a tropical moduli space. We show that this tropical moduli space is a refinement of the moduli space of spin tropical curves. We describe explicitly the induced decomposition of its cones. This is a joint work with Abreu and Pacini.

--XX--XX--

Speaker 2: Victor Pretti (UNICAMP)

Title: Rank 0 Asymptotic Bridgeland stability

Abstract: Bridgeland stability is a modern tool to study stability of objetcs in triangulated categories, and specially in the derived category of coherent sheaves over a smooth projective variety. Its asymptotic version, as studied by Bridgeland, Bayer and Jardim--Maciocia, is known to behave like Gieseker stability for sheaves in various situations. In this seminar we will focus on Bridgeland stabilities over the projective space P^3 and its asymptotic behaviour for rank zero objects to prove their respective relation with Gieseker stability for sheaves.

Two 30 min presentations by PhD students or recently graduated PhDs.

Cancelled
2021-04-07 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/41/
by Cancelled as part of Brazilian algebraic geometry seminar

Abstract
With great sadness, we cancel the BRAG seminar today (07 April 2021) due to the passing of our dear friend and colleague Roberto Callejas Bedregal, last night by covid. May he always be remembered for his joy, generosity and good heart, and may he rest in peace.

Aline Andrade (UFF)
2021-04-14 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/42/
Title: On rank 3 instanton bundles on projective 3 space (https://stable.researchseminars.org/talk/BRAG/42/)
by Aline Andrade (UFF) as part of Brazilian algebraic geometry seminar

Abstract
We investigate rank $3$ instanton bundles on $\mathbb{P}^3$ of charge $n$ and its correspondence with rational curves of degree $n+3$. in order to prove that the generic stable rank 3 ’t Hooft bundle of charge n is a smooth point in the moduli space of rank 3 vector bundles of Chern classes (0,n,0). Additionally, for $n=2$ we present a correspondence between stable rank $3$ instanton bundles and stable rank $2$ reflexive linear sheaves and we prove that the moduli space of rank $3$ stable locally free sheaves on $\mathbb{P}^3$ of Chern classes $(0,2,0)$ is irreducible, generically smooth of dimension 16. (Joint work with D. R. Santiago, D. D. Silva, and L. S. Sobral)

Javier Ribon (UFF)
2021-05-05 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/43/
Title: Local foliations with closed leaves (https://stable.researchseminars.org/talk/BRAG/43/)
by Javier Ribon (UFF) as part of Brazilian algebraic geometry seminar

Abstract
Our goal is describing the leaf space of local holomorphic foliation whose leaves are closed in a neighborhood of the origin. By considering the holonomy groups associated to leaves, it is necessary to study the finitely generated groups of local biholomorphisms whose orbits are closed (or equivalently finite) in some neighborhood of the origin. We show that for such groups, there is always an analytic curve through the origin that is contained in the fixed point set of a finite index subgroup. Moreover, we outline the properties of the linear part of a local diffeomorphism with finite orbits. The result provides a stability result à la Reeb in intermediate dimension for dimension one foliations defined in the neighborhood of a compact invariant curve. This is a joint work with Lucivanio Lisboa and some of the results are part of his PhD thesis.

Elizaveta Vishnyakova (UFMG)
2021-05-12 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/44/
Title: Donagi-Witten construction and a graded covering of a supermanifold (https://stable.researchseminars.org/talk/BRAG/44/)
by Elizaveta Vishnyakova (UFMG) as part of Brazilian algebraic geometry seminar

Abstract
In the paper "Super Atiyah classes and obstructions to splitting of
supermoduli space", Donagi and Witten suggested a construction of a
first obstruction class for splitting of a supermanifold via
differential operators. We generalize this idea. As a result we
obtain a family of embeddings of the category of supermanifolds into
the category of iterated vector bundles and into the category of
graded manifolds. It was shown that the images of a supermanifold in
these categories satisfy universal properties of a graded covering or
a graded semicovering. In our talk we will discuss these functors in the case of a Lie supergroup and a Lie superalgebra. (Joint work with M. Rotkiewicz).

João Paulo Figueiredo (IMPA)
2021-04-28 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/45/
Title: Regular foliations on rationally connected threefolds with nef anticanonical bundle (https://stable.researchseminars.org/talk/BRAG/45/)
by João Paulo Figueiredo (IMPA) as part of Brazilian algebraic geometry seminar

Abstract
In his classification of regular foliations on surfaces, Brunella showed that every regular foliations on a rational surface is algebraically integrable, with rational leaves. This leads to the conjecture, due to Touzet, that every regular foliation on a rationally connected manifold is algebraically integrable with rationally connected leaves. This conjecture was shown to be true by Druel for the case of Fano manifolds. In this talk, we will present progress towards this conjecture for threefolds, by showing that it is true for regular foliations of codimension one on threefolds with nef anticanonical bundle.

Emilio Franco (IST Lisboa)
2021-05-19 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/46/
Title: Deformation theory for orthogonal and symplectic sheaves (https://stable.researchseminars.org/talk/BRAG/46/)
by Emilio Franco (IST Lisboa) as part of Brazilian algebraic geometry seminar

Abstract
Moduli spaces of principal bundles usually carry interesting
geometric structures, being a powerful, and often unique, source of
examples of varieties with prescribed properties and characteristics.
Nevertheless, these spaces might be non-compact whenever the base
(smooth) scheme has dimension higher than 1. Principal sheaves provide a
natural compactification of the moduli space of principal bundles for a
connected complex reductive structure group. Therefore, moduli spaces of
principal sheaves are projective varieties equipped with an interesting
geometry, at least, on a dense subset. In order to check whether or not
these properties extend to the compactification, we need a local
description of the moduli spaces, precisely over the locus where the
principal sheaves fail to be principal bundles. Such description would
naturally derive from deformation theory of principal sheaves, which is
still missing at present date.

In this talk we consider orthogonal and symplectic sheaves, and show
that the deformation and obstruction theory of these objects is
controlled by a deformation complex naturally built out of our starting
orthogonal (resp. symplectic) sheaf.

Valeriano Lanza (UFF)
2021-05-26 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/47/
Title: Moduli of flags of sheaves: a quiver description (https://stable.researchseminars.org/talk/BRAG/47/)
by Valeriano Lanza (UFF) as part of Brazilian algebraic geometry seminar

Abstract
In [1], the moduli spaces of framed flags of sheaves on P2 were described by means of representations of the so-called enhanced ADHM quiver. We first review those results, with a recent refinement concerning the chamber structure in the space of stability parameters. We shall then discuss the obstruction theory for these moduli spaces, showing in general that they have a perfect obstruction theory, and providing for specific choices of invariants a class of unobstructed points. Finally, open problems and possible further developments will be presented. This is a joint work with Rodrigo von Flach and Marcos Jardim.

[1] R. A. von Flach and M. Jardim, Moduli spaces of framed flags of sheaves on the projective plane. Journal of Geometry and Physics 118 (2017), 138–168.

Alex Massarenti (U. Ferrara)
2021-04-21 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/48/
Title: Complete symplectic quadrics and Kontsevich moduli spaces of conics in Lagrangian Grassmannians (https://stable.researchseminars.org/talk/BRAG/48/)
by Alex Massarenti (U. Ferrara) as part of Brazilian algebraic geometry seminar

Abstract
Given a reductive algebraic group $G$ and a Borel subgroup $B$, a spherical variety
is a normal variety admitting an action of $G$ with an open dense $B$-orbit. A special
class of spherical varieties are the so-called wonderful varieties. These are smooth
spherical varieties for which we require $G$ to be semisimple and simply connected
and the existence of an open $B$-orbit whose complementary set is a simple normal
crossing divisor. We will construct the wonderful compactification of the space of
symmetric, symplectic matrices on which the symplectic group acts. Furthermore,
we will compute the Picard group of this compactification and we will study its
birational geometry in low-dimensional cases. As an application, we will recover the
results on the birational geometry of the Kontsevich spaces of conics in Grassmannians
due to I. Coskun and D. Chen, and we will prove new results on the birational
geometry of the Kontsevich spaces of conics in Lagrangian Grassmannians.
This is a joint work with Elsa Corniani.

William Montoya (UNICAMP)
2021-06-02 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/49/
Title: An extension of the Noether-Lefschetz loci in toric varieties (https://stable.researchseminars.org/talk/BRAG/49/)
by William Montoya (UNICAMP) as part of Brazilian algebraic geometry seminar

Abstract
In this talk, I will motivate and prove a Noether-Lefschetz type theorem for quasi-smooth intersections in a projective simplicial toric variety with suitable conditions, which implies that the Hodge conjecture holds on a very general quasi-smooth intersection variety and that the natural extension of the Noether-Lefchetz loci is not empty. The Noether-Lefschetz loci can be understood as the loci where the Hodge conjecture is unknown. If time allows me, I will also show that the Hodge conjecture is also true for some varieties in the Noether-Lefschetz loci. This is joint work with Prof. Ugo Bruzzo.

Abramo Hefez, Steven Kleiman, Michelle Morgado, Marcelo Saia, José Seade, Otoniel Nogueira da Silva
2021-06-09 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/50/
Title: An afternoon of algebraic geometry in memory of Roberto Bedregal (https://stable.researchseminars.org/talk/BRAG/50/)
by Abramo Hefez, Steven Kleiman, Michelle Morgado, Marcelo Saia, José Seade, Otoniel Nogueira da Silva as part of Brazilian algebraic geometry seminar

Abstract
we convene for a special edition of the BRAG (Brazilian Algebraic Geometry Seminar) session in honor of our friend and colleague Roberto Callejas-Bedregal, who passed away on 6 April of this year. We will have 15 minute lectures given by Abramo Hefez (UFF), Steven Kleiman (MIT), Michelle Morgado (Unesp), Marcelo José Saia (UFSCar), José Seade (UNAM) and Otoniel Nogueira da Silva (UNAM), along with testimonials from Roberto's students, friends and colleagues.

-- xx -- xx --

1 - Abramo Hefez: The mathematical trajectory of Roberto Bedregal Abstract:
The aim of this short talk is to sketch Bedregal's journey towards his mathematical achievements.

-- xx -- xx --

2 - Steve Kleiman: Roberto's MIT Thesis
To begin, some introductory remarks are made about how Roberto
came to study at MIT and to develop the first purely algebraic treatment
of the Whitney Conditions. Then the main result of his thesis is
stated, and its terms, explained. Finally, an inkling is given of the
import of the Whitney Conditions.

-- xx -- xx --

3 - José Seade: Chern classes for singular varieties.
I will speak about some of the joint work I did recently with Roberto Callejas-Bedregal and Michelle Morgado, about Chern classes of singular varieties.

-- xx -- xx --

4 - Michelle Morgado: Lê cycles

-- xx -- xx --

5 - Marcelo José Saia: On Segre numbers of homogeneous map germs

-- xx -- xx --

6 - Otoniel Nogueira da Silva: Equisingularity of families of map germs and Ruas's Conjecture

-- xx -- xx --

7 - Presentation of a video made by Roberto's students.

Gaia Comaschi (University of Campinas)
2021-06-16 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/51/
Title: Instanton sheaves of low charge on Fano threefolds (https://stable.researchseminars.org/talk/BRAG/51/)
by Gaia Comaschi (University of Campinas) as part of Brazilian algebraic geometry seminar

Abstract
Let $X$ be a Fano threefold of Picard number one and of index $2+h, \ h=0,1$.
An \textit{instanton sheaf of charge $k$ on $X$} is defined as a semi-stable rank 2 torsion free sheaf $F$ with Chern classes $c_1=-h, \ c_2=k, \ c_3=0$ and such that $F(-1)$ has no cohomology.
Locally free instantons, originally defined on the projective space and later generalised on other Fano threefolds $X$, had been largely studied from several authors in the past years; their moduli spaces present an extremely rich geometry and useful applications to the study of curves on $X$.
In this talk I will illustrate several features of non-locally free instantons of low charge on 3 dimensional quadrics and cubics. I will focus in particular on the role that they play in the study of the Gieseker-Maruyama moduli space $M_X(2;-h,k,0)$ and describe how we can still relate these sheaves to curves on $X$.

Marcos Jardim (University of Campinas)
2021-06-23 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/52/
Title: Logarithmic sheaves for complete intersections (https://stable.researchseminars.org/talk/BRAG/52/)
by Marcos Jardim (University of Campinas) as part of Brazilian algebraic geometry seminar

Abstract: TBA

Letterio Gatto (Politecnico di Torino)
2021-06-30 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/53/
Title: On the Vertex Operators Representation of Lie Algebras of Matrices (https://stable.researchseminars.org/talk/BRAG/53/)
by Letterio Gatto (Politecnico di Torino) as part of Brazilian algebraic geometry seminar

Abstract
Relying on our common skill to multiply square matrices by vectors, recalled in the first part of the talk, we will describe the exterior algebra of a polynomial ring in one indeterminate, and/or the ring of symmetric polynomials, as a representation of the Lie algebra of matrices of infinite size with all but finitely many zero entries.

The description is achieved by bridging classical Schubert calculus on Grassmannians to the vertex operators occurring in the so-called boson-fermion correspondence (Poincaré duality for Grassmannians of infinite dimensional linear spaces) and highlights a substantial generalization of a classical picture drawn in the Eighties by Date, Jimbo, Kashiwara and Miwa, within the framework of algebraic analysis and infinite dimensional completely integrable systems. The talk will survey joint work with (in alphabetical order) O. Behzad, A. Contiero, D. Martins, P. Salehyan, I. Scherbak and R. Vidal Martins.

Arturo Fernández Pérez (Federal University of Minas Gerais (UFMG))
2021-08-25 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/55/
Title: Number of Milnor and Tjurina of Foliations (https://stable.researchseminars.org/talk/BRAG/55/)
by Arturo Fernández Pérez (Federal University of Minas Gerais (UFMG)) as part of Brazilian algebraic geometry seminar

Abstract
In this talk, I will show the relationship between the Milnor and Tjurina numbers of a foliation in the complex plane. Such numbers are similar to the classic Milnor and Tjurina numbers for singular curves. This work is in collaboration with Evelia García Barroso (Universidad de la Laguna - Spain) and Nancy Saravia Molina (PUCP-Peru)

Thiago Fassarella (UFF)
2021-09-01 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/56/
Title: Moduli spaces of parabolic Higgs bundles (https://stable.researchseminars.org/talk/BRAG/56/)
by Thiago Fassarella (UFF) as part of Brazilian algebraic geometry seminar

Abstract
We are interested in studying moduli spaces of parabolic Higgs bundles on (punctured) curves. A ramified cover between curves defines a map between the corresponding moduli spaces, and we will discuss the behavior of this map with respect to the Hitchin fibration. This is joint work in progress with Frank Loray.

Hossein Movasati (IMPA)
2021-09-08 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/57/
Title: Ibiporanga: A moduli space for differential equations of automorphic forms (https://stable.researchseminars.org/talk/BRAG/57/)
by Hossein Movasati (IMPA) as part of Brazilian algebraic geometry seminar

Abstract: TBA

Rodrigo Salomão (UFF)
2021-09-15 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/58/
Title: On the classification of fibrations by singular curves on unirational surfaces (https://stable.researchseminars.org/talk/BRAG/58/)
by Rodrigo Salomão (UFF) as part of Brazilian algebraic geometry seminar

Abstract
In 1944 Zariski discovered that Bertini’s theorem on variable singular points is no longer true when we pass from a field of characteristic zero to a field of positive characteristic. In other words, he found fibrations by singular curves, which only exist in positive characteristic. Such fibrations are connected with many interesting phenomena. For instance, the extension of Enrique’s classification of surfaces to positive characteristic (Bombieri and Mumford in 1976), the counterexamples of Kodaira vanishing theorem (Mukai in 2013 and Zheng in 2016) and the isolated singularities with infinity Milnor number (jointly work with Hefez and Rodrigues in 2019). In this talk we are going to show that the smoothing process introduced by Shimada in 1991 can be used to describe the set of fibrations by genus two singular curves on unirational surfaces, up to isomorphism among their generic fibers, such that the smoothing are elliptic fibrations. Moreover we will also describe the vector fields whose tangencies with elliptic fibrations generate such fibrations by singular curves, after the quotient of the rational elliptic surfaces. This is a work in progress with J. H. O. Rodrigues and R. O. C. Santos.

Ommolbanin Behzad / Eduardo Vital (University of Isfahan, Iran / IMPA, Brazil)
2021-09-29 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/59/
Title: Exterior powers, Polynomial rings and Representation of Lie Algebras / Degenerations of linear series to curves with three components, using quiver representations (https://stable.researchseminars.org/talk/BRAG/59/)
by Ommolbanin Behzad / Eduardo Vital (University of Isfahan, Iran / IMPA, Brazil) as part of Brazilian algebraic geometry seminar

Abstract
Young BRAG with two short presentations:

Speaker 1: Ommolbanin Behzad (University of Isfahan, Iran)

Title: Exterior powers, Polynomial rings and Representation of Lie Algebras

Abstract: I will report on some recent work of myself, A. Contiero, D.
Martins, R. Vidal Martins about representing lie algebras of vector space
endomorphisms on exterior algebras, seeing it as the finite type case of the
celebrated DJKM bosonic vertex operator representation of gl∞(Q).

-- xx -- xx --

Speaker 2: Eduardo Vital (IMPA, Brazil)

Title: Degenerations of linear series to curves with three components, using quiver representations

Abstract: We explore the existence of simple bases for certain special quiver representations arising from degenerations of linear series on nodal curves. The existence
of a simple basis implies that the representation decomposes into representations
of dimension one and simplifies the calculus of the Hilbert polynomial of the
quiver Grassmannian associated to the representation. For these quiver representations, we characterise the existence of a simple basis with a local condition.
And to a nodal curve with three components we show that its linked projective
space is Cohen-Macaulay, reduced, and has pure dimension.
This is a joint work in progress with Eduardo Esteves and Renan Santos.

Jean Vallès (University of Pau, France)
2021-10-06 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/60/
Title: Free curves obtained from a pencil of algebraic curves (https://stable.researchseminars.org/talk/BRAG/60/)
by Jean Vallès (University of Pau, France) as part of Brazilian algebraic geometry seminar

Abstract
An algebraic curve of the projective plane is free if its module of tangent vectors fields is free,
in other words, if it is generated globally by two tangent vector fields.
Being given two algebraic curves without common component I propose a new method to produce free curves
and a theorem explaining why they are free.

Jacqueline Arancibia (UFPB)
2021-10-27 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/61/
Title: Some applications of Bott's Localization Formula in Enumerative Problems (https://stable.researchseminars.org/talk/BRAG/61/)
by Jacqueline Arancibia (UFPB) as part of Brazilian algebraic geometry seminar

Abstract
First of all we explain what are the mathematical objects involved in the Bott's localization formula. Next, we apply the Bott's localization formula to resolve the classical problem of four lines (from Schubert's Calculus), as well as, to compute the degree of the subvariety formed by the surfaces of degree d in ${\mathbb P}^3$ containing a conic and two points varying on a fixed line.

Cleto B. Miranda-Neto (UFPB)
2021-11-03 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/62/
Title: Vanishing of Ext modules and characterizations of smoothness in algebraic varieties (https://stable.researchseminars.org/talk/BRAG/62/)
by Cleto B. Miranda-Neto (UFPB) as part of Brazilian algebraic geometry seminar

Abstract
In this talk, after presenting some homological results which establish freeness criteria for modules, I will discuss how the vanishing of suitable Ext modules can be translated in terms of smoothness of a complex algebraic variety at a given point. Special attention will be given, e.g., to complete intersections and rational surface singularities.

Hamid Hassanzadeh (UFRJ)
2021-11-10 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/63/
Title: A genus explanation of Buchsbaum-Rim multiplicity (https://stable.researchseminars.org/talk/BRAG/63/)
by Hamid Hassanzadeh (UFRJ) as part of Brazilian algebraic geometry seminar

Abstract
We explain the Buchsbaum-Rim multiplicity as the Euler-Poincare characteristic of an “ordinary” Koszul complex. This provides a generalization of Serre’s formula for the Hilbert-Samuel multiplicity in terms of the length of the Koszul homologies.The talk is based on a joint work with: Viniçius Bouça, Thiago Fiel and Jose Naeliton.

Cristhian Garay (CIMAT, Guanajuato)
2021-11-17 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/64/
Title: Inflection polynomials of linear series on superelliptic curves (https://stable.researchseminars.org/talk/BRAG/64/)
by Cristhian Garay (CIMAT, Guanajuato) as part of Brazilian algebraic geometry seminar

Abstract
We explore the inflectionary behavior of linear series on families of marked superelliptic curves (i.e., cyclic covers of P^1). The inflection of these linear series supported away from the superelliptic ramification locus is parameterized by the inflection polynomials, a certain infinite class of polynomials generalizing the division polynomials (which are used to compute the torsion points of elliptic curves).

These polynomials are remarkable since their properties reflect aspects of the underlying family of superelliptic curves. We also obtain inflectionary varieties, which describe the global behaviour of the inflection points on the family.

In this talk we will introduce these inflection polynomials and some of their properties. We report on joint work with Ethan Cotterill, Ignacio Darago, Changho Han, and Tony Shaska.

Margarida Melo (Roma Tre University)
2021-11-24 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/65/
Title: Tropicalization of the universal Jacobian: logarithmic and non-archimedean viewpoints (https://stable.researchseminars.org/talk/BRAG/65/)
by Margarida Melo (Roma Tre University) as part of Brazilian algebraic geometry seminar

Abstract
We construct a tropical universal Jacobian on the category of cone stacks, generalizing previous work by Abreu and Pacini. We then show that this cone stack can be obtained by tropicalizing two versions of the algebraic universal Jacobian: the logarithmic universal Jacobian and the non-archimedean universal Jacobian. We then discuss the connection between our construction and Molcho-Wise’s logarithmic Picard and Picard stacks. The talk will be based on joint work with S. Molcho, M. Ulirsch, F. Viviani and J. Wise.

https://impa-br.zoom.us/j/86824790044?pwd=[REDACTED]

Eduardo Esteves (IMPA)
2021-12-01 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/66/
Title: Degenerations of line bundles and divisors along curves (https://stable.researchseminars.org/talk/BRAG/66/)
by Eduardo Esteves (IMPA) as part of Brazilian algebraic geometry seminar

Abstract
A family of line bundles along a family of smooth curves parameterized by the
punctured disk can be extended in several ways over the limit stable curve of the
family. On the other hand, the associated divisors can each be extended in a unique way. We will discuss the interplay between these two dynamics, part of an ongoing work to address the problem raised by Eisenbud and Harris in the 1980’s of constructing a useful moduli of limit linear series over the
moduli of stable curves. It involves collaboration with Piere Rodríguez, Renan Santos, Eduardo Vital (IMPA) and Omid Amini (École Polytechnique).

Renato Vidal Martins (UFMG)
2021-12-08 18:30 - 20:00 | https://stable.researchseminars.org/talk/BRAG/67/
Title: The Gonality of an Integral Curve (https://stable.researchseminars.org/talk/BRAG/67/)
by Renato Vidal Martins (UFMG) as part of Brazilian algebraic geometry seminar

Abstract
This talk introduces gonality for arbitrary integral curves, and reports
on joint work in progress with Steve Kleiman. Discussed are topics that
appear naturally, including the following: linear series defined by
torsion-free sheaves of rank 1; the relation among gonality, scrolls,
and canonical models; and an upper bound on gonality.

Carolina Araujo (Impa)
2023-04-26 17:30 - 18:30 | https://stable.researchseminars.org/talk/BRAG/68/
Title: The Calabi Problem for Fano Threefolds (https://stable.researchseminars.org/talk/BRAG/68/)
by Carolina Araujo (Impa) as part of Brazilian algebraic geometry seminar

Abstract
The Calabi Problem is a formidable problem in the confluence of differential and algebraic geometry. It asks which compact complex manifolds admit a Kähler-Einstein metric. A necessary condition for the existence of such a metric is that the canonical class of the manifold has a definite sign. For manifolds with zero or positive canonical class, the Calabi problem was solved by Yau and Aubin/Yau in the 1970s. They confirmed Calabi's prediction, showing that these manifolds always admit a Kähler-Einstein metric. On the other hand, for projective manifolds with negative canonical class, called “Fano manifolds”, the problem is much more subtle: Fano manifolds may or may not admit a Kähler-Einstein metric. The Calabi problem for Fano manifolds has attracted much attention in the last decades, resulting in the famous Yau-Tian-Donaldson conjecture. The conjecture, which is now a theorem, states that a Fano manifold admits a Kähler-Einstein metric if and only if it satisfies a sophisticated algebro-geometric condition, called “K-polystability”. In the last few years, tools from birational geometry have been used with great success to investigate K-polystability. In this talk, I will present an overview of the Calabi problem, the recent connections with birational geometry, and the current state of the art in dimension 3.

Thiago Fassarella (UFF)
2023-04-26 19:00 - 20:00 | https://stable.researchseminars.org/talk/BRAG/69/
Title: Torelli-type theorems (https://stable.researchseminars.org/talk/BRAG/69/)
by Thiago Fassarella (UFF) as part of Brazilian algebraic geometry seminar

Abstract
In this talk, I will present some Torelli-type theorems concerning moduli spaces of parabolic vector bundles and logarithmic connections over an elliptic curve. In the case of vector bundles, we extend a classical theorem of D. Mumford and P. Newstead to the parabolic context (joint work with Luana Justo). For connections, we recover the spectral data from the symplectic structure of the moduli space (joint work with Frank Loray and Alan Muniz).

Israel Vainsencher (UFMG)
2023-06-29 14:00 - 14:50 | https://stable.researchseminars.org/talk/BRAG/70/
Title: Grau de componentes do espaço de folheações em P^3 (https://stable.researchseminars.org/talk/BRAG/70/)
by Israel Vainsencher (UFMG) as part of Brazilian algebraic geometry seminar

Abstract
Revisitaremos algumas componentes conhecidas, e.g.,
1) $\{aA dB-bB dA, a=\deg A/mdc, b=\deg B/mdc\}$;
2) pullback linear $\mathbb{P}^3 \rightarrow \mathbb{P}^2$;
3) pullback linear $\mathbb{P}^3 \rightarrow \mathbb{P}^1 \times \mathbb{P}^1$.
O primeiro caso tem grau conhecido apenas se $a|b$ ou se $a=2$ e $b$ é ímpar. O segundo caso admite uma fórmula explícita. Por fim, explicaremos trabalho em andamento com Vivi Ferrer sobre o grau de componentes descobertas recentemente por Wodson Mendson.

Mauricio Corrêa (U. Bari)
2023-06-29 15:00 - 15:50 | https://stable.researchseminars.org/talk/BRAG/71/
Title: Enumerative geometry of Legendrian foliations: a tale of contact (https://stable.researchseminars.org/talk/BRAG/71/)
by Mauricio Corrêa (U. Bari) as part of Brazilian algebraic geometry seminar

Abstract
A foliation by curves is called Legendrian if it is tangent to some distribution of contact on a projective 3-space. Our goal is to give formulas for the dimensions and degrees of the varieties of Legendrian foliations, and of the varieties of foliations tangent to a pencil of planes which are in terms of Athus polynomial. This is a joint work with Israel Vainsencher.
