Colóquio de Geometria e Aritmética
Rio de Janeiro
Descrição: Colóquio rotativo entre IMPA, UFRJ e UFF. Acontecerá toda última sexta feira do mês, contando com duas palestras, uma de 10:30 as 11:30 e outra de 12:00 as 13:00. As palestras serão ministradas por pesquisadores locais e do exterior das áreas de geometria algébrica e aritmética.
O objetivo principal é estimular a troca de ideias e a colaboração entre as instituições do Rio de Janeiro. A pausa para o café e o almoço após o colóquio darão oportunidade para discussões entre os participantes e interação com os palestrantes.
Esperamos uma grande participação dos pesquisadores da área. Estudantes de mestrado e doutorado são encorajados a participar.
24/8, UFF. Campus do Gragoatá, Bloco H, sala 407.
Misha Verbitsky (IMPA),
Transcendental Hodge algebra|
Resumo. Let M be a projective manifold. Transcendental Hodge lattice of weight p is the smallest rational Hodge substructure in Hp(M) containing Hp,0(M). Transcendental Hodge lattice is a birational invariant of M. Yu. Zarhin computed the transcendental Hodge lattice for a K3 surface, and proved that it is an irreducible representation of a unitary or orthogonal group over a number field. I will prove that the direct sum of all transcendental Hodge lattices for any projective manifold is an algebra, and compute it (using Zarhin's theorem) explicitly for all hyperkahler manifolds in terms of
irreducible representations of unitary or orthogonal groups over number fields.
Omid Amini (ENS, Paris),
Trees in algebraic geometry|
Resumo. The aim of the talk is to show through examples of recent results how combinatorial objects called trees naturally arise and play a role in understanding the geometry of algebraic varieties and their asymptotics. The results themselves are motivated by arithmetic geometry and mathematical physics.
28/9, IMPA, sala 236.
Luciane Quoos (UFRJ),
Resumo. Let Fq be the finite field with q elements. A polynomials f(x) in Fq[x] is said to be a permutation polynomial (PP for short) if it is a bijection as a function in Fq. In this talk it will be presented some criterions to decide when a polynomial over Fq is a PP and some families that can be obtained using these criterions. The notion of the Carlitz Rank of a PP will be introduced and a bound involving its Carlitz Rank, the degree of a certain polynomial and the cardinality of the finite field will be given.
Kostiantyn Iusenko (USP),
Stable representations of posets.|
Resumo. Representations of finite dimensional algebras can be approached combinatorially via representations of posets (due to L.A. Nazarova
and A.V. Roiter) and representations of quivers (due to P. Gabriel). The problem of classifying representations of "most" algebras is wild in a sense
that it is as difficult as the problem of classifying representations of free algebras. Nevertheless, one can use geometrical approach by considering the spaces whose points correspond naturally to isomorphism classes of representations. Using standard GIT methods A. King defined the moduli spaces of quiver representations. In this talk we will discuss certain aspects related to study of moduli space of poset representations. We will see that the Euler quadratic form associated with a poset plays significant role here: for calculation of dimension of moduli space and for canonical choice of stability (which is certain analogue of Schofield's characterization of Schurian roots for quiver). Also we plan to discuss the behavior of Coxeter transformations on stable representations.
26/10, UFRJ, sala C116.
Alex Massarenti (UFF),
On Mori chamber and stable base locus decompositions|
Resumo. The effective cone of a Mori dream space admits two wall-and-chamber decompositions called Mori chamber and stable base locus decompositions. In general the former is a non trivial refinement of the latter. We will investigate the differences between these decompositions for moduli spaces of complete collineations. Furthermore, we will provide a criterion to establish whether the two decompositions coincide for a Mori dream space of Picard rank two, and we will construct an explicit example of a Mori dream space of Picard rank two for which the decompositions are different, showing that our criterion is sharp.
Edgar Costa (MIT),
Resumo. In this talk, we will focus on how one can deduce some geometric invariants of an abelian variety or a K3 surface by studying their Frobenius polynomials.
In the case of an abelian variety, we show how to obtain the decomposition of the endomorphism algebra, the corresponding dimensions, and centers.
Similarly, by studying the variation of the geometric Picard rank, we obtain a sufficient criterion for the existence of infinitely many rational curves on a K3 surface of even geometric Picard rank.
30/11, UFF. Campus do Gragoatá, Bloco H, sala 407.
Jean Vallès (Universitá de Pau et de Pays de L'adour),
Logarithmic bundles and line arrangements, an approach via the standard construction|
Resumo. We propose an approach to study logarithmic sheaves associated with hyperplane arrangements on the projective space, based on projective duality, direct image functors and vector bundles methods. We focus on free line arrangements, recovering, thanks to this new approach, many known results and improving some of them about the so-called Terao's conjecture.
Giancarlo Urzúa (PUC Chile),
On explicit boundedness for surfaces|
Resumo. Kollár and Shepherd-Barron (1988) introduced a natural compactification to the Gieseker moduli space of surfaces of general type with fixed invariants K2 and χ, which is analogous to the Deligne-Mumford (1969) compactification of the moduli space of curves of genus g>1. This compactification is coarsely represented by a projective scheme (due to Kollár 1990) because of Alexeev's proof of boundedness (1994). Surfaces parametrized by this moduli space are called (KSBA) stable surfaces. They admit at most slc singularities, and they may or may not be degenerations of canonical projective surfaces of general type (i.e. at most ADE singularities and K ample). Since this KSBA moduli space is represented by a projective scheme, we have a finite list of slc singularities in the stable surfaces which it parametrizes. It is a hard problem to write down that list.
In this talk, I will show effective bounds for a particular but relevant class of singularities (T-singularities) in stable surfaces W with fixed K2. When W is not rational, we can classify surfaces attaining the bound. When W is rational we can show where the problem is (to find optimal bounds). This is a joint work with Julie Rana. Similar bounds were found by Jonny Evans and Ivan Smith via symplectic topology, which can be seen as an obstruction to embed symplectically a rational homology ball Bp,q in a canonically polarized surface. At the end, I will show that non-canonically embedded Bp,q's are unbounded in surfaces of general type, so the condition in Evans-Smith was necessary, by means of the explicit stable 3-dimensional MMP for T-degenerations (joint with Paul Hacking and Jenia Tevelev). This last result is joint with Jonny Evans.
14/12, IMPA, sala 236.
Giosuè Muratore (UFMG),
Betti numbers and pseudoeffective cones in 2-Fano varieties|
Resumo. In the first part, we study the 2-Fano varieties. Defined by De Jong and Starr, they satisfy some higher dimensional analogous properties of Fano varieties. We propose a definition of (weak) k-Fano variety and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties in analogy with the case k=1. Then, we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index greater or equal to n-2, and also we complete the classification of weak 2-Fano varieties of Araujo and Castravet.
In the second part we study the indeterminacy locus of the Voisin map. Beauville and Donagi proved that the variety of lines F(Y) of a smooth cubic fourfold Y is a hyperkähler variety. Recently, C. Lehn, M.Lehn, Sorger and van Straten proved that one can naturally associate a hypekähler variety Z(Y) to the variety of twisted cubics on Y. Then, Voisin defined a degree 6 rational map: f:F(Y)xF(Y)-->Z(Y). We will show that the indeterminacy locus of f is the locus of intersecting lines.
Claudia Polini (University of Notre Dame - USA),
Bounding degrees of vector fields|
Resumo. For the purpose of deciding whether an algebraic vector field in the complex projective plane is algebraically integrable, Poincaré asked whether the degree of any algebraic curve left invariant by such a vector field can be bounded above in terms of the degree of the vector field. Although the question has a negative answer in general, it has inspired a great deal of work for over a century. A broader goal of this research is to relate the degree of vector fields to properties of curves that they leave invariant.
We will report on recent joint work with Marc Chardin, Hamid Hassanzadeh, Aron Simis, and Bernd Ulrich where the question is approached from a more algebraic point of view. We provide lower bounds for the degree of vector fields in terms of local and global data of the curves they leave invariant. The sharpness of the bounds will be discussed.
Nivaldo Medeiros (UFF)
Carolina Araujo (IMPA)
Cecília Salgado (UFRJ)
Eduardo Esteves (IMPA)
Marco Pacini (UFF)
Oliver Lorscheid (IMPA)
Apoio: CNPq, Capes, Faperj
Contato: colga at impa.br