Colóquio de Geometria e Aritmética
Rio de Janeiro


Programa: 2017.2

1/9, IMPA. Auditorio 1

10:30-11:30 Hamid Hassanzadeh (UFRJ), Effective criteria for graded and bigraded birational maps
Resumo. In this talk, we consider rational maps whose source is a product of two subvarieties, each one being embedded in a projective space. Our main objective is to investigate birationality criteria for such maps. First, a general criterion is given in terms of the rank of a couple of matrices that became to be known as Jacobian dual matrices. This criterion generalizes a previous criterion for projective varieties. By mention the applications and motivation, then, we focus on rational maps from P1 x P1 to P2 in very low bidegrees and provide new matrix-based birationality criteria by analyzing the syzygies of the defining equations of the map.
This talk is based on a Joint work with L. Buse, N. Botbol, M.Chardin, T.Hoa and A. Simis
12:00-13:00 Ugo Bruzzo (SISSA, Trieste), The Noether-Lefschetz problem for normal threefolds.
Resumo. The classical Noether-Lefschetz problem is about the Picard number of surfaces in P3. For d greater or equal 4, if S(d) is the locus of smooth surfaces in the linear system O(d) on P3, the generic surface in S(d) has Picard number one. It is possible to estimate the codimensions of the components of S(d), and show that their codimension is related to geometric properties of the surfaces they describe.
Over the last few years, in collaboration with A. Grassi, and more recently with A.F. Lopez, I have investigated generalizations to surfaces in normal, Q-factorial 3-folds. My talk will be devoted to describe some of the results we obtained.

29/9, UFF. Campus do Gragoatá, Bloco H, sala 407 (como chegar)

10:30-11:30 Gaël Cousin (IMPA), Algebraizable isomonodromic deformations.
Resumo. The talk concerns isomonodromic deformations of logarithmic connections on a complex projective curve C. After some motivational discussion, we will explain how one can translate the algebraizability of the universal isomonodromic deformation in terms of the monodromy representation r of the initial connection. Namely, (the conjugacy class of) r has finite orbit under the mapping class group of the punctured curve if and only if the deformation is algebraizable.
The main arguments are Riemann-Hilbert correspondence and a topological construction. A byproduct of this work is a tool to construct regular flat connections on varieties fibered by curves. This is a joint work with Viktoria Heu.
12:00-13:00 Inder Kaur (IMPA), Existence of semistable vector bundles of fixed rank and determinant on fibred surfaces in mixed characteristic.
Resumo. It is well-known that the moduli space of semistable vector bundles of fixed rank and degree on a surface may be empty. In this talk I will prove the existence of a semistable vector bundle of fixed rank and determinant on a fibred surface defined over a Henselian discrete valuation ring (of mixed characteristic) under the assumption that the special fibre is a semistable generalised tree-like curve.

27/10, UFRJ. Sala C116 Bloco C

10:30-11:30 Vladimir Mitankin, Integral points on generalised affine Châtelet surfaces.
Resumo. A generalised affine Châtelet surface over the integers is defined by y^2 - az^2 = P(t), where a is a non-zero integer and P(t) is a separable polynomial with integral coefficients. Building up on an earlier work of Colliot-Thélène and Sansuc which suggests the use of Schinzel's hypothesis we show that the integral Brauer-Manin obstruction is the only obstruction to the integral Hasse principle for a family of such surfaces. We do so by injecting tools from algebraic number theory. Moreover, when there is no integral Brauer--Manin obstruction we show that the set of integral points on any surface in the family satisfies a strong approximation property in t away from infinity.
12:00-13:00 Javier Fernandez de Bobadilla, Reflexive modules on gorenstein surface singularities.
Resumo. We generalize clasical constructions of Artin, Verdier, Esnault, Wunram, and Khan concerning Mckay correspondence to arbitrary Gorenstein surface singularities. We study also the relevant deformation theory.
This is joint work with Agustín Romano.

1/12, IMPA. Sala 336

10:30-11:30 Thiago Fassarela (UFF), Conexões sobre curvas elípticas.
Resumo. Nesta palestra vamos introduzir o espaço de moduli de conex&etildees logarítmicas com n polos e com resíduos fixados sobre uma curva de gênero g. Tal espaço admite uma estrutura de variedade algébrica, simplética de dimensão 2N, onde N=3g-3+n. Pretendemos descrever esta estrutura no caso n=2 e g=1. Em particular obtemos um resultado do tipo Torelli que diz que levando em consideração a estrutura simplética, podemos recuperar os dados iniciais: a curva com dois pontos marcados, o divisor de polos e os resíduos. Trabalho com conjunto com Frank Loray.
12:00-13:00 Alexandre Fernandes (UFC), O grau de conjuntos complexos afim como invariante geométrico.
Resumo. Abordaremos a invariância do grau de conjuntos complexos afim por transformações bi-Lipschitz.

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