Colóquio de Geometria e Aritmética
Rio de Janeiro


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Programa: 2015.1

6/3, IMPA. Sala 228

10:30-11:30 Julie Deserti (Institut de Mathématiques de Jussieu, França), Some properties of the Cremona groups.
Resumo. I will give some properties of the 2-dimensional Cremona group and explain why we don't have such properties in higher dimensions.
12:00-13:00 Laura Costa (Universitat de Barcelona, Espanha), Ulrich bundles on Grassmannians.
Resumo. The existence of Ulrich bundles (i.e. bundles without intermediate cohomology whose corresponding module has the maximal number of generators) on a projective variety is a challenging problem with a long and interesting history behind and few known examples. The goal of this talk is to give a full classification of all homogeneous Ulrich bundles on a Grassmannian $Gr(k, n)$ of $k$-planes on $P^n$. To end, we will determine the representation type of the Grassmannians.

29/4, IMPA, sala 232. Nota que este dia é uma quarta-feira.

10:30-11:30 Sándor Kovács, Positivity and moduli.
Resumo. The ultimate aim of this lecture is to explain recent results, joint with Zsolt Patakfalvi, on projectivity of moduli spaces of stable log-varieties of general type and on Iitaka's conjecture regarding subadditivity of log-Kodaira dimension for fiber spaces. On route to discussing these new results I will review background material on moduli spaces and positivity as well as previous results on projectivity of moduli spaces and on Iitaka's conjecture. The central tools of the proofs are a strengthening of Kollár's celebrated Ampleness Lemma and its application to pushforwards of relative pluricanonical sheaves and their determinants. Time permitting I will explain the essentials of this approach without getting into technical details.
12:00-13:00 Sandra Di Rocco, Toric vector bundles and polytopes.
Resumo. We will motivate and define the category of toric vector bundles over smooth complete toric varieties. After highlighting some applications and potential applications, we will concentrate on extending various properties of line bundles on smooth toric varieties to higher-rank toric vector bundles. Klaychko’s characterization of toric vector bundles intrinsically carries the definition of associated polytopes, extending the corresponding theory of line bundles. Positivity properties, as global generation and generation of jets, are visualised in terms of convex properties of the corresponding polytopes and provide useful criteria. I will illustrate this correspondence and show how these criteria lead to proving and disproving connections between various notions of positivity and cohomology vanishing. This is joint work with G. Smith and K. Jabbusch.

29/5, IMPA. Esse coloquiuo faz parte do workshop Rational Points.

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