Colóquio de Geometria e Aritmética
Rio de Janeiro


Programa: 2016.1

29/4, IMPA. Auditorio 1

10:30-11:30 Roberto Bedregal (UFPB), On Local Cohomology Modules with Arbitrary Support.
Resumo. Let R be a commutative Noetherian ring with unity. If M is an R-module and I is an ideal of R, the i-th local cohomology module of M with support in I plays an important role in Commutative Algebra and Algebraic Geometry. In this talk I will give a quick overview on the properties these modules satisfy and show how we generalize them to local cohomology modules with arbitrary support. This is a joint work with L. Alba-Sarria and N. Caro-Tuesta.
12:00-13:00 Enrique Arrondo Esteban (Madrid), On a conjecture of Hartshorne
Resumo. A subvariety of dimension n in the N-dimensional projective space is necessarily determined by at least N-n equations, and only in special cases can be determined by exactly N-n equations (if this happens, the subvariety is called a complete intersection). Hartshorne conjectured that, when the subvariety is smooth and n>2N/3, then it is a complete intersection. The smoothnes assumption implies that, at least in a local neighborhood of any point, the subvariaty can be defined by N-n local equations. A first natural question is when such local equations glue to produce a section of a vector bundle of rank N-n over the projective space. The second natural question is when a vector bundle over the projective space splits as a direct sum of line bundles. A complete answer to those questions would produce an answer to Hartshorne's conjecture. Our talk, which intends to be elementary, will consist of an overview of those problems.

3/6, UFRJ. Sala C 116

10:30-11:30 Jethro van Ekeren (UFF), Indicators of Heisenberg Groups.
Resumo. Consider a module M over some type of algebra for which the notion of `dual module' makes sense (e.g., compact groups, Lie algebras, vertex algebras, etc). To say that M is self-dual is the same as to give an invariant bilinear form on M. Schur's lemma implies at once that such a form is either symmetric or else skew-symmetric, and one defines the Frobenius-Schur (FS) indicator of M to be +1 or -1 in these respective cases.
The motivation for this talk comes from vertex algebras, where calculation of FS indicators is (or has been) a notoriously delicate and slippery matter. The actual content of the talk, however, will have to do with finite Heisenberg groups: I will present a new and elementary calculation of some FS indicators in this context using finite quadratic spaces and the 2-adic lattice genus. Aside from streamlining the most technical part of several existing vertex algebra constructions, this calculation suggests the start of a nice story involving lattice automorphisms, Weil representations, arithmetic, and braided tensor categories.
12:00-13:00 Simone Marchesi (UNICAMP), The existence of monads on projective varieties.
Resumo. We will generalize Floystad's criterion for the existence of monads and we will study the properties of the sheafs defined by their cohomology. In particular we will study whenever it is locally free, simple, stable and we will describe the set of pairs of morphisms which define the monad and the moduli space of the cohomology sheaves.
This is a joint work with Pedro Macias Marques and Helena Soares.

24/6, UFF. Campus do Gragoatá, Bloco H, sala 407 (como de chegar)

10:30-11:30 Hossein Movasati (IMPA), A historical introduction to Hodge theory
Resumo. The origin of Hodge theory goes back to many works on elliptic, abelian and multiple integrals. In this talk I am going to explain how Lefschetz was puzzled with the computation of Picard rank and this lead him to consider the homology classes of curves inside surfaces. This ultimately was formulated in Lefschetz (1,1) theorem and then the Hodge conjecture.
12:00-13:00 Felipe Voloch (Canterbury), Geradores de curvas eliticas sobre corpos finitos
Resumo. Consideraremos o problema de determinar geradores para o grupo de pontos racionais de curvas eliticas sobre corpos finitos. Mostraremos como construir um conjunto pequeno contendo geradores. Tambem discutiremos porque não há uma formula geral para os geradores e a relação disso com as conjecturas de Artin e Lang-Trotter. Finalmente, discutiremos uma conjectura de Poonen e sua relação com a construção de pontos de ordem grande.

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