Colóquio de Geometria e Aritmética
Rio de Janeiro


Programa: 2013.2

30/08, UFF. Sala de seminários da pós-graduação, 7o. andar

10:30-11:30 Roi do Campo (UFF), Jacobian discrepancies and rational singularities
Resumo. In this talk I will introduce the notion of Jacobian discrepancy, an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. This invariant, very natural from the point of view of jet schemes and arc spaces, leads to a framework in which adjunction and inverstion of adjunction hold in full generality. Moreover, they allow us to give explicit formulas measuring the gap between the dualizing sheaf and the Grauert-Riemenschneider canonical sheaf of a normal variety, leading to characterizations of rational singularities in terms of discrepancies.
12:00-13:00 Filippo Viviani (Roma Tre), Fourier-Mukai transform and autoduality for compactified Jacobians
Resumo. Compactified Jacobians of singular curves with planar singularities appear as fibers of the Hitchin map for the moduli space of Higgs bundles. The classic limit of the geometric Langlands conjecture (as formulated by Donagi-Pantev) predicts a Fourier-Mukai autoequivalence of the fibers of the Hitchin map. For reduced curves, we are able to define a Poincar? sheaf and to show that the induced integral transform defines a Fourier-Mukai autoequivalence of any fine compactified Jacobian. This was previously establish for integral curves by D. Arinkin. As a corollary, we establish an autoduality result for compactified Jacobians, generalizing previous results of Arinkin, Esteves, Gagn e, Kleiman, Rocha. The major novelty for reducible curves is that there are many (possibly non-isomorphic) fine compactified Jacobians, all of which are birational Calabi-Yau projective varieties with mild singularities. More generally, we show that any two such compactified Jacobians are derived equivalent. This is in line with Kawamata's conjecture that birational Calabi-Yau (smooth) varieties should be derived equivalent and it seems to suggest an extension of this conjecture to (mildly) singular Calabi-Yau varieties. This is a joint work with M. Melo and A. Rapagnetta.

27/09, UFRJ. Sala: C 116

10:30-11:30 Jérémy Blanc (Basel), Dynamical degrees of birational transformations of surfaces
Resumo. The dynamical degree ?( f ) of a rational transformation f measures the exponential growth rate of the degree of the formulae that define the n-th iterate of f. We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of ?( f ) and the structure of the conjugacy class of f. For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed, well ordered set of algebraic numbers. Joint work with Serge Cantat.
12:00-13:00 Ivan Pan (Montevideo), On the Jonquieres type maps of the projective space
Resumo. We consider the set of birational maps of the dimension 3 projective space which stabilize a net of lines passing through a point. We note that such a set is a subgroup, G say, of the so-called Cremona group which is not finitely generated. We show that G is generated by 8 well-known involutions together with elements in a subgroup J of G. Next we give a homological characterization of elements in J. Joint work with Aron Simis.

25/10, IMPA. Sala: 232

10:30-11:30 Severino Collier (UFRJ), Foliations of projective space without invariant proper algebraic subvarieties of positive dimension.
Resumo. We construct families of one dimensional foliations over n-dimensional projective space whose generic fibre has no proper invariant subvariety of positive dimension and whose parameter space has dimension linear in n and independent of the degree of the foliation.
12:00-13:00 Sam Payne (Yale), Nonarchimedean methods for multiplication maps
Resumo. Multiplication maps on linear series are among the most basic structures in algebraic geometry, encoding, for instance, the product structure on the graded pieces of the homogeneous coordinate ring of a projective variety. In this talk, I will discuss joint work with Dave Jensen, developing tropical and nonarchimedean analytic methods for studying multiplication maps of linear series on algebraic curves in terms of piecewise linear functions on graphs, with a view toward applications in classical complex algebraic geometry.

29/11, UFF. Sala: Sala de seminários da pós-graduação, 7o. andar

10:30-11:30 Benoît Claudon (CNRS/IMPA), Algebraicity of universal cover of projective varieties
Resumo. In this talk we will investigate the following question: which are the smooth projective varieties whose universal cover can be endowed with a quasi-projective structure? In particular, we will see that quite surprisingly this question is related to the Abundance Conjecture and that its validity implies a complete description of such varieties. This is based on a joint work with Andreas Höring and János Kollár.
12:00-13:00 Julio Bueno de Andrade (IHES), Random Matrices, L-functions and Hyperelliptic Curves.
Resumo. In this seminar I will present some connections between Random Matrix Theory and the theory of Zeta functions of curves over finite fields. Specifically I will describe the Katz-Sarnak philosophy for the study of mean values of Zeta functions of curves

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