Abstracts

• Dan Avritzer, Desargues configurations via polar conics of plane cubics
  Dados dois triângulos no plano complexo A1B1C1 e A2B2C2, o teorema clássico de Desargues afirma que se as retas A1A2, B1B2 e C1C2 se encontram em um ponto A então o prolongamento dos lados correspondentes A1B1 e A2B2, A1C1 e A2C2, B1C1 e B2C2 se encontram em pontos alinhados. O conjunto dos 10 pontos e retas do teorema é conhecido como uma configuração de Desargues. No princípio do século passado grande foi o interesse que este teorema despertou. O matemático grego Stephanos mostrou como associar à uma configuração de Desargues, uma séxtica binária. Posteriormente Marr estudou como associar à uma séxtica binária, uma configuração de Desargues. Nesta conferência revisaremos os resultados no contexto atual da teoria geométrica dos invariantes mostrando que os dois mapas são inversos um do outro.
  

• Carles Bivià-Ausina, Mixed multiplicities of ideals and the integral closure of modules
  Let R denote the ring of formal power series with coefficients in the field of complex numbers. We characterize the class of submodules M of the free module R^p whose integral closure splits as a direct sum of complete ideals. One of the characterizations we give is expressed in terms of the Buchsbaum-Rim multiplicity of M and the mixed multiplicities of some ideals attached to M, whenever M is a
  submodule of R^p of finite colength.
  
  
• Roberto Bedregal, Álgebras Multigraduadas e Multiplicidades de Buchsbaum-Rim generalizadas
  O fecho integral de um ideal é um objeto algébrico que tem tido muitas aplicações em varios aspectos de geometria algébrica e álgebra comutativa. Além de ter sido estudado per se na teoria multiplicativa de ideais, ele tem sido uma ferramenta fundamental na solução de problemas da teoria de singularidades. De fato, para estudar as condições de Whitney, tem-se que entender o fecho integral do produto do ideal Jacobiano e o ideal maximal de A. Por outro lado, Rees [R] provou no seu famoso teorema, que atualmente leva seu nome, que dois ideais m-primários J \subset I de um anel local formalmente equidimensional têm o mesmo fecho integral se, e somente se, suas multiplicidades coincidem.
  Para estender o Teorema de Rees para ideais arbitrários de um anel local (A,m) é necessário estender a noção de multiplicidade de Hilbert-Samuel para estes ideais. Isto foi feito por R. Achilles e M. Manaresi em [AM]. Eles definiram, para cada ideal I de um anel d-dimensional (A,m), uma sequência de multiplicidades c_0(I,A), ..., c_d(I,A). Esta sequência satisfaz a propriedade de que se o ideal I é m-primário então c_0(I,A) é a multiplicidade de Hilbert-Samuel de I e é o único elemento não nulo de esta sequência. Além disso, R. Achilles e S. Rams [AR] provaram que esta sequência coincide com a sequência formada pelos números de Segre do ideal I, como definidas por [GaG]. Por outro lado, Gafiney e Gassler provaram, usando métodos geométricos, que dois ideais tem o mesmo fecho integral se, e somente se, tem a mesma sequência de números de Segre. Em resumo temos o seguinte resultado válido no contexto analítico:
  
  Theorem 1.1 Seja A=O_{X,0} o anel local de uma variedade analítica equidimensional X. Se J \subset I são ideais arbitrários de A, então \bar{I}=\bar{J} se, e somente se, c_k(I,A) =c_k(J,A) para todo k= 0,...,d.
  
  Neste trabalho pretendemos proporcionar uma demonstração puramente algébrica deste resultado, assim como definir as multiplicidades mistas para ideais arbitrários que generalizem as multiplicidades mistas de Teissier para ideais m-primários.
  Por outro lado, D. Buchsbaum e D. Rim [BR] introduziram a noção de multiplicidade de um módulo M, denotada por e_{BR}(M), onde M é um submódulo próprio de um módulo livre E de posto p, tal que o quociente E/M tenha comprimento finito. Esta multiplicidade, que leva seus nomes, coincide com a multiplicidade de Hilbert-Samuel no caso em que M é um ideal de A. Estas multiplicidades satisfazem tambem o Teorema de Rees, no sentido que se N \subset M \subset E tem co-comprimento finito então N é uma redução de M se, e somente se, e_{BR}(N) =e_{BR}(M). O objetivo principal deste trabalho é o de introduzir uma sequência de multiplicidades c_0(M,A), ..., c_{d+p-1}(M,A), associadas a um submódulo próprio M de um módulo livre E de posto p, que coincida com a squência de Achilles-Manaresi no caso idealístico e que generalize também a multiplicidade de Buchsbaum-Rim, no sentido que se M tem co-comprimento finito então c_0(M) é a multiplicidade de Buchsbaum-Rim de M e é o único elemento não nulo desta sequência. Além disso, esta sequência deve também satisfazer o Teorema de Rees para módulos:
  
  Theorem 1.2 Seja (A,m) um anel local d-dimensional e N \subset M \subset E submódulos próprios de um módulo livre E de posto p. Então, N é uma redução de M se, e somente se, c_k(N) =c_k(M) para todo k= 0,...,d+p-1.
  
  Referências:
  [AM] Achilles, R. and Manaresi, M., Multiplicities of a bigraded ring and intersection theory, Math. Ann.309, (1997) 573-591.
  [AR] Achilles, R. and Rams, S., Intersection numbers, Segre numbers and generalized Samuel multiplicities, Arch. Math.77, (2001) 391-398.
  [BR] Buchsbaum, D. and Rim, D. Rim, A generalized Koszul complex II. Depth and Multiplicity, Trans. Amer. Math. Soc.111, (1964) 197-224.
  [GaG] Gafiney, T. and Gassler, R., Segre numbers and hypersurface singularities, J. Algebraic Geom.8 (1999), no. 4, 695-736.
  [R] Rees, D., a-transform of local rings and a theorem on multiplicities of ideals, Proc. Cambridge Philos Soc. (2) 57 (1961) 8-17.
  
  
• Fernando Cukierman, Infinitesimal deformations of integrable foliations
  We study the Zariski tangent space to the moduli space of integrable algebraic foliations, mainly of codimension one, in a complex projective space. Applying this to the problem of determining the irreducible components of the moduli space, we obtain alternative proofs of some known theorems and exhibit some new irreducible components, consisting of foliations with split tangent sheaf, which includes examples of foliations induced by group actions. (Joint work with Jorge Vitório Pereira)
  
  
• Eduardo Esteves, The compactified Jacobian of reducible curves
  I will give an overview of certain constructions of compactifications of relative Jacobians over families of reducible curves. Some questions and open problems will be presented and discussed.
  
  
• Arnaldo Garcia, On the Galois closure of towers
  We show that the Galois closure of certain towers of function fields over finite fields is asymptotically good.We then apply our method to some of the towers in the literature, showing that some Galois closures have the same limits as the original towers (in one case it attains the Drinfeld-Vladut upper bound, and in the other case it attains the lower bound of T.Zink).
  
  
• Letterio Gatto, Intersection formulas in Grassmann Varieties
  I will quickly report on some joint work with Taise Santiago, about general combinatorial formulas for intersection of Schubert cycles. Degrees formulas (in top codimension) will be also discussed: they generalize well known ones for the degree of grassmannians in the Plücker embedding.
  
  
• Hemar Godinho, Conditions for the solvability of systems of two and three additive forms over p-adic fields
  During this seminar we are going to present some results regarding non-trivial solvability in p-adic integers of systems of two and three additive forms. Assuming that the congruence equation ax^k+by^k+cz^k \equiv d (mod p) has a solution with xyz \not\equiv (mod p) we have proved that any system of two additive forms of odd degree k with at least 6k+1 variables, and any system of three additive forms
  of odd degree k with at least 14k+1 variables has always non-trivial p-adic solutions, provided p does not divide k. The assumption of the solubility of the congruence equation above is guaranteed for example if p>k⁴.
  In the particular case of degree k=5 we have proved the following results. Any system of two additive forms with at least n variables has always non-trivial p-adic solutions provided n≥31 and p>101 or n≥36 and p>11. Furthermore any system of three additive forms with at least n variables has always non-trivial p-adic solutions provided n≥61 and p>101 or n≥71 and p>11.

• Barry Green, Counting finite order automorphisms of the p-adic open disc
  In this talk we report on recent results on the fixed point geometry of order p automorphisms of the p-adic disc, the associated semistable models as well as work in progress of those of higher p-power order. In particular we discuss how these results determine the number of such automorphisms up to change of parameters in special cases.
  
  
• Cem Güneri, Additive Equations and Weights in Cyclic Codes
  Weights of codewords in a cyclic code over Fq are intimately related to number of solutions of Artin-Schreier equations over an extension Fq^m of Fq. This was used by J. Wolfmann to give a general minimum distance bound on cyclic codes. However, Wolfmann's bound is restricted to those cyclic codes for which the related family of equations are irreducible over Fq^m, since he uses Weil bound to estimate the number of solutions. By giving an explicit factorization for a class of reducible Artin-Schreier type equations, we obtain a Weil type bound for their number of solutions and also extend Wolfmann's result.
  If time permits, we shall also discuss the extension of this idea to multidimensional cyclic codes. This time, weights of codewords can be related to family of curves or higher dimensional hypersurfaces. All of the results to be presented are obtained together with F. Özbudak.
  
  
• Abramo Hefez, An overview on the analytic classification of plane branches
  In this talk we will show how to solve the classification problem of plane analytic branches and the corresponding moduli problem, posed by Zariski in the seventies. This is joint work with M. E. Hernandes.
  
  
• Marcelo Escudeiro Hernandes, An Algorithm for the Analytic Classification of Plane Branches
  We will present an algorithm that performs the analytic classification of plane branches. Given a plane branch, we compute the set L of natural numbers which are the values of Kähler differentials of the curve with respect to the valuation determined by it. This set L is an analytic invariant for plane branches. Associated to each set L, there is a normal form which is analytically equivalent to the original plane branch.
  Two normal forms are equivalent if and only if they differ by an homothety (a very special coordinate change). It is very simple to check whether two such normal forms are or not equivalent.
  
  
• Marc Hindry, Why is it difficult to compute the Mordell-Weil group?
  The Mordell-Weil group is the finitely generated group of rational points of an elliptic curve or abelian variety over a number field. We want to discuss an analog of the Brauer-Siegel (which states that the class number times the regulator of units of a number field has roughly the size of the square root of the discriminant). The analog is a conjecture stating that the order of the Tate-Shafarevic group times the regulator of an abelian variety should have roughly the size of the exponential of the Faltings height of the abelian variety.
  
  

• Marcos Jardim, ADHM construction of instanton sheaves on CPⁿ
  I will outline the ADHM construction of instanton sheaves on P² and show how it can be generalized to provide a construction of torsion-free instanton sheaves on Pⁿ of trivial splitting type, leading to a nice description of moduli space of these objects.
  
  
• Hiren Marahaj, Explicit constructions of algebraic-geometric codes from towers
  There are many explicit constructions of recursively defined asymptotically good towers of function fields. The first few levels of these towers provide excellent examples of curves with many points for code construction. Since the code length from towers increase exponentially with the level, only codes from the first few levels are expected to be of current practical interest. In this talk, we present a simple technique to produce easily described bases for a class of high performance codes from the first few levels of any given recursively defined tower.
  
  
• Nivaldo Medeiros, On Wronskian classes in the Chow ring of the moduli of curves
  We shall outline a systematic method to express certain Wronskian classes in the Chow ring of the moduli space of stable curves of genus g, in terms of the usual generators. For example, when g=3, our method recovers the well-known expression for the closure of the locus of hyperelliptic smooth curves, without resorting to the ad hoc method of test curves.This is a joint work with E. Esteves and L. Gatto.
  
  
• Massimiliano Mella, Birational geometry of uniruled 3-folds: hopes and fears
  I will outline a tentative project to better understand the birational geometry of uniruled 3-folds. This is based on the "experimental data" collected in recent years using the theory of Maximal singularities and Sarkisov Program. Warning: there will be many more questions than answers!
  
  
• Ferruh Özbudak, On a class of function fields over finite fields
  We study a class of function fields over finite fields and we obtain existence and characterization results for some function fields with prescribed genera and number of rational places in this class. In particular we show the existence of minimal and maximal function fields in the class, we show the maximal function fields in the class are Galois subcovers of the corresponding Hermitian function field and we determine the corresponding Galois group. This is a joint work with Emrah Çakçak.
  

• Amílcar Pacheco, Abelian Lehmer problem for Drinfeld modules
  Let A=Fq[t], k= Fq(t), K a finite extension of k and Kab the maximal abelian extension of K. Let Φ be a Drinfeld A-module of generic characteristic defined over K. We show that the canonical height of an element in Kab is bounded below by a constant which just depends on Φ and K. This is joint work with S. David.
   
  
• Jorge Vitório Pereira, The Poncelet-Legendre Transform
  I will describe the Poncelet-Legendre Transform for webs on the projective plane. Emphasis will be given in the correspondence between degree 3 foliations and 3-webs of degree 1.
  
  
• Francesco Russo, On special Cremona transformations
  
  Any Cremona transformation defined by a homaloidal system of primals with a single irreducible non-singular base variety is a rare enough phenomenon to merit special study.
  
  J.G. Semple - J.A. Tyrrell (1968)
  
  We shall present the classification of some special Cremona transformations, the class defined above by Semple and Tyrrell. Describing the geometrical restrictions imposed to the base locus schemes, together with the results obtained for those defined by quadratic primals, we shall try to explain why only very few examples are known and why the exceptional ones can be, in principle, completely classified.
  
  
• Marcelo Saia, On the Zariski conjecture of the multiplicity for topologically equivalent hypersurfaces
  (Joint work with J. N. Tomazella)
  Zariski asked in the article Open questions in the Theory of Singularities, Bull. of the A.M.S. 1971, the following question:
  
  Whether for a hypersurface singularity the multiplicity is an invariant of the topological type?

  A positive answer for this question was previously known in the case of plane curves, for homogeneous surfaces and for quasi homogeneous complex hypersurfaces with isolated singularity, as shown by Greuel in Constant Milnor number implies constant multiplicity for quasihomogeneous singularities, Manuscripta Math., 1986. Trotman gives a positive answer for Zariski's question for the case of first order deformations of type F(x,t) = f(x)+tg(x) of a complex germ f with isolated singularity in his Thesis, Equisingularité et conditions de Whitney, Orsay, France, 1980.
  In this talk we discuss this question for the case of any deformation of type F(x,t) = f(x) +\sum_{s=1}^{p}t^sg_s(x), where f has isolated singularity at 0. We give some examples to ilustrate how to work with this kind of families and conclude that Zariski' s question also has a positive answer for these families.

  
  

• Aron Simis,  Birational Combinatorics
  Birational combinatorics is the theory of characteristic-free rational maps between projective spaces defined by monomials. A facet of the theory will be developed, along with natural criteria for such maps to be birational onto their image. The main focus is set on the underlying combinatorial elements rather than on special algebraic properties of the base ideal (base locus), a method that has been emphasized in recent years. Finally, one shows a bridge between the method of syzygies and Rees algebra and the present combinatorial one, by way of comparing the respective linear algebra gadgets - from modules over the ground polynomial ring over a field to modules over the integers.
  
  
  

• Israel Vainsencher,  Degrees of some components of the space of foliations
  Let P, Q be homogeneous polynomials of degrees p, q. The logarithmic differential form w=pPdQ - qQdP defines a codimension 1 foliation in Pⁿ, provided gcd(P,Q)=1. Such foliations form a component of the space of foliations. We calculate the degree of this component, assuming that either gcd(p,q)=1 or p divides q.