Seminari de Geometria Algebraica de Barcelona
 UB UPC UAB
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 Seminari de Geometria Algebraica 2017/2018 Conferenciant Títol Data i hora Eduard Casas-Alvero Universitat de Barcelona On the analytic classification of irreducible plane curve singularities I will present new results regarding which Puiseux coefficients the analytic type of a complex irreducible plane curve singularity depends on. Divendres 29 de setembre, 15h, Aula T2, FMI-UB Ignasi Mundet Universitat de Barcelona Symplectic finite group actions on $$\Large{S^2\times T^2}$$ Let $$X=S^2\times T^2$$. For any symplectic form $$\omega$$ of $$X$$ we denote by $$Symp(X,\omega)$$ the group of symplectomorphisms of $$(X,\omega)$$. In this talk we will explain different results on the finite subgroups of $$Symp(X,\omega)$$. The main results are: for any $$\omega$$ there exist infinitely many isomorphism classes of finite subgroups of $$Diff(X)$$ which are not represented by any finite subgroup of $$Symp(X,\omega)$$; for any $$\omega$$ there exists another symplectic form $$\omega'$$ and a finite subgroup of $$Symp(X,\omega')$$ which is not isomorphic to any finite subgroup of $$Symp(X,\omega)$$. We will sketch the proofs, which use the theory of pseudoholomorphic curves. We will make an effort to make the talk understandable without previous knowledge of symplectic geometry. Divendres 6 d'octubre, 15h, Aula T2, FMI-UB Gianfranco Casnati Politecnico di Torino, Itàlia Ulrich bundles on some classes of surfaces in projective spaces An Ulrich bundle on a variety in the projective space is a vector bundle whose associated module of sections has a linear resolution over the projective space. Ulrich bundles have many interesting properties and their existence on a fixed variety has several geometric consequences for it. Ulrich bundles on curves can be easily described. This is no longer true for Ulrich bundles on a surface, though many results are known. In the talk we focus our attention on this latter case proving the existence of Ulrich bundles on some classes of surfaces, giving some results on the size of the families of Ulrich bundles on them and sometimes dealing with their stability properties. Divendres 27 d'octubre, 15h, Aula T2, FMI-UB Fatmanur Yıldırım INRIA Sophia-Antipolis & UB Finite fibers of multi-graded rational maps on $$\Large{\mathbb{P}^3}$$ I will present a new method to study the fibers of a rational multi-graded map $$\Psi$$ from $$\mathbb{P}^2\times\mathbb{P}^1$$ (or $$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$$) to $$\mathbb{P}^3$$, which is a joint work with Nicolás Botbol, Laurent Busé and Marc Chardin. My motivation is to compute the distance from a point $$p\in\mathbb{R}^3$$ to an algebraic rational surface $$\mathcal{S}\in\mathbb{R}^3$$. Firstly, from a parametrization of $$\mathcal{S}$$, I will construct a homogeneous parametrization $$\Psi$$ for the normal lines to $$\mathcal{S}$$, where $$\Psi$$ is a multi-graded rational map from $$\mathbb{P}^2\times\mathbb{P}^1$$ (or $$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$$) to $$\mathbb{P}^3$$. Then, I will describe the fibers over a point $$p\in\mathbb{P}^3$$. After that, I will state a matrix $$\mathcal{M}(\Psi)_{\nu}$$ of a certain multi-degree $$\nu$$ obtained by the syzygies of ideal generated by the coordinates of $$\Psi$$. Divendres 3 de novembre, 15h, Aula T2, FMI-UB Meritxell Saez Cornellana University of Copenhagen,Dinamarca Positive solutions to linear systems slides Usually in applications, where variables represent measurable quantities, only nonnegative solutions are meaningful. Hence, criteria to decide the positivitity of the solutions to a system of equations are desired. I will present some of the known results in this direction and a new criteria for linear systems based on a multidigraph associated with the equations. The main motivation for this work has been on the application to biochemical reaction networks that I will briefly present. Divendres 10 de novembre, 15h, Aula T2, FMI-UB Jose Ignacio Burgos Gil ICMAT (CSIC) Where do little elliptic curves go? Let C be a curve over $$\mathbb{Q}$$ provided with an integral model, an ample line bundle on the model and a semipositive metric. To these data we can associate the height of the curve and the height of every algebraic point of the curve. The essential minimum of the curve is the minimal accumulation point of the height of the algebraic points. The essential minimum is a mysterious and elusive invariant. A result of Zhang shows that the essential minimum has a lower bound in terms of the height of the curve, and an example of Zagier shows that there can be several isolated values of the height below the essential minimum. When C is the modular curve, the line bundle agrees with the bundle of modular forms and the metric is the Weil-Petersson metric, then the height of an algebraic point agrees with the stable Faltings height of the corresponding elliptic curve. In this talk we will discuss methods of proving lower and upper bounds for the essential minimum and apply them to the modular curve, giving a partial description of the spectrum of the stable Faltings height of elliptic curves. This is joint with with Ricardo Menares and Juan Rivera-Letelier. Divendres 17 de novembre, 15h, Aula T2, FMI-UB Ana Belén de Felipe Universitat de Barcelona Topology of spaces of valuations and geometry of singularities Given an algebraic variety X defined over a field k, the space of all valuations of the field of rational functions of X extending the trivial valuation on k is a projective limit of algebraic varieties. This space had an important role in the program of Zariski for the proof of the existence of resolution of singularities. In this talk we will consider the subspace RZ(X,x) consisting of those valuations which are centered in a given closed point x of X and we will focus on the topology of this space. In particular we will concentrate on the relation between its homeomorphism type and the local geometry of X at x. We will characterize this homeomorphism type for regular points and normal surface singularities. This will be done by studying the relation between RZ(X,x) and the normalized non-Archimedean link of x in X coming from the point of view of Berkovich geometry. Divendres 24 de novembre, 15h, Aula T2, FMI-UB Julio José Moyano Fernández Universitat Jaume I de Castelló Some instances of generating series in algebraic geometry The use of zeta functions has a long tradition in algebraic geometry and related fields, coming back to the classical works of Riemann, Dedekind or Hilbert. They offer an elegant way to encode many invariants of the typical objects under consideration. In this talk we will focus on certain series defined in local contexts, namely the so-called Poincaré series attached to curve singularities. We will describe the relation of those Poincaré series with some zeta functions coming from a rather number-theoretical setting, and shed some light on the role that they play in global contexts. The talk will contain some pieces of collaborations with F. Delgado, A. Melle and W. Zúñiga. Divendres 1 de desembre, 15h, Aula T2, FMI-UB Matthias Nickel Goethe Universität Frankfurt am Main, Alemanya Newton-Okounkov bodies of exceptional curve valuations Newton-Okounkov bodies are convex bodies associated to line bundles on projective varieties that capture positivity properties of the line bundle in question. Let p be a closed point in $${\mathbb{P}}_{\mathbb{C}}^2$$ and consider a surface obtained by a sequence of finitely many blowups of points where we start with p and always blow up a point in the exceptional divisor created last. Our result is that the Newton-Okounkov body of the pullback of $$\mathcal{O} _{\mathbb{P}^2}(1)$$ with respect to the flag given by the last exceptional divisor and a point on it is either a triangle or a quadrilateral. Furthermore a characterization of both cases can be given using the dual graph and the supraminimal curve. This is joint work with Carlos Galindo, Francisco Monserrat and Julio José Moyano-Fernández. Divendres 15 de desembre, 15h, Aula de l'IMUB, FMI-UB Simone Marchesi IMECC - UNICAMP, Brasil Line arrangements and vector bundles One of the most famous and interesting conjectures regarding line arrangements (we will restrict to the projective plane case) is the so called Terao's conjecture, which basically states that the freeness of an arrangement depends on its combinatorics. If the conjecture is not true, than the arrangement would be associated to a vector bundle whose jumping locus is related to a 0-dimensional scheme in the projective plane. In this talk we will focus on the case when such scheme is a point, characterizing the associated vector bundles and relating, through examples, this jumping point to the line arrangement. Divendres 12 de gener, 15h, Aula T2, FMI-UB David Marín Universitat Autònoma de Barcelona Foliations and webs with continuous symmetries on complex projective surfaces We will describe the structure of foliations and webs on complex projective surfaces which are invariant by a germ of birational flow. We will discuss in detail the case of the projective plane, characterizing planar projective webs with many infinitesimal symmetries. This is a joint work with Marcel Nicolau. Divendres 19 de gener, 15h, Aula T2, FMI-UB Antonio Macchia Università degli Studi di Bari,Itàlia Slack ideals of Polytopesslides We introduce a canonical realization space for a polytope that arises as the positive part of a real variety. The variety is determined by the so-called slack ideal of the polytope, that encodes its combinatorics. The slack ideal offers new ways to categorize polytopes in terms of complexity. Our constructions provide a uniform computational framework for several classical questions about polytopes such as rational realizability, projectively uniqueness, non-prescribability of faces, and realizability of combinatorial polytopes. The simplest slack ideals are toric. We identify the toric ideals that arise from projectively unique polytopes: all d-polytopes with d+2 facets or vertices have such slack ideals but there are more interesting examples in this set. We illuminate the relationship between projective uniqueness and toric slack ideals using new and classical examples. (Joint work with João Gouveia, Rekha Thomas and Amy Wiebe) Divendres 26 de gener, 15h, Aula T2, FMI-UB Workshop on Complex Algebraic Geometry - Pirola 60th 5 - 9 de febrer 2018, Aula B5, UB Alessandro Oneto Universitat Politècnica de Catalunya On the Hilbert function of general fat points in $$\Large{\mathbb{P}^1\times\mathbb{P}^1}$$slides Polynomial interpolation problems have been largely studied in algebraic geometry and commutative algebra. The classical question is the following: how many independent conditions a general union of fat points in the projective space $$\mathbb{P}^n$$ give on the complete linear system of hypersurfaces of given degree? The case of double points has a very long history which goes back to the classical school of algebraic geoemetry of XIX century, but a complete solution has been given by J. Alexander and A. Hirschowitz in 1995, after a series of enlightening papers where they introduced to powerful méthode d'Horace différentiel. For higher multiplicities, even the case of planar curves is in general open. A conjectural answer in this case is given the so-called SHGH Conjecture (due to B. Segre, B. Harbourne, A. Gimigliano and A. Hirschowitz) In this talk, we consider a multi-graded version of the classical question. We take ideals defining schemes of fat points (with same multiplicity and generic support) in $$\mathbb{P}^1\times\mathbb{P}^1$$ and we want to compute how many independent conditions they impose on the linear system of curves of given bi-degree. In 2005, M.V. Catalisano, A.V. Geramita and A. Gimigliano introduced a so-called multiprojective-affine-projective method that reduce this problem to the standard graded case of fat points in $$\mathbb{P}^2$$. In their work, they completely solve the case of double points in $$\mathbb{P}^1\times\mathbb{P}^1$$. After an historical introduction and after explaining the Horace method and the multiprojective-affine-projective method, I will present a joint work with M.V. Catalisano and E. Carlini where we use these methods to give a complete answer in the case of triple points in $$\mathbb{P}^1\times\mathbb{P}^1$$ ( arXiv: 1 711.06193). Partial results for higher multiplicity will be also presented. Divendres 16 de febrer, 15h, Aula T2, FMI-UB Francesco Strazzanti INdAM - Universitat de Barcelona Binomial edge ideals of bipartite graphs In the last decades the connections between commutative algebra and combinatorics have been extensively explored. In general it is interesting to study classes of ideals in a polynomial ring by associating with them combinatorial objects, such as simplicial complexes, graphs, clutters or polytopes. In this talk we are interested in the so-called binomial edge ideals, which are ideals generated by binomials corresponding to the edges of a finite simple graph. They can be viewed as a generalization of the ideal of $2$-minors of a generic matrix with two rows. After an introduction to these ideals, we will provide a classification of Cohen-Macaulay binomial edge ideals of bipartite graphs by giving an explicit construction in graph-theoretical terms. To prove this classification we will make use of the dual graph of an ideal, showing in our setting the converse of the Hartshorne's Connectedness theorem. This is a joint work with Davide Bolognini and Antonio Macchia. Divendres 2 de març, 15h, Aula T2, FMI-UB Jorge Martín Morales Universidad de Zaragoza Resolving some surface singularities with weighted blow-upsslides In some cases the general algorithm for resolving a surface singularity is not very efficient in practice, since it often appears too many divisors which do not contribute to the topology of the singularity. In the talk a special type of toric blow-ups will be introduced so as to resolve certain surface singularities. Divendres 9 de març, 15h, Aula T2, FMI-UB José Ángel González Prieto Universidad Complutense de Madrid Topological Quantum Field Theories and their application to Hodge theory slides Topological Quantum Field Theories are powerful categorical tools that provide deep insight into the behaviour of topological invariants. In this talk, we will discuss some properties of TQFTs and we will give a general construction procedure as a combination of a field theory and a quantisation. Using this method, we will construct a lax monoidal TQFT that computes the mixed Hodge structure on the cohomology of representation varieties. In this business, Saito's mixed Hodge modules will play an important role as quantizations of Hodge structures. Joint work with M. Logares and V. Muñoz. Divendres 6 d'abril, 15h, Aula T2, FMI-UB Enrico Sbarra Università degli Studi di Pisa,Itàlia Jet schemes and determinantal varieties Jet schemes and arc spaces received quite a lot of attention by researchers after their introduction, due to J. Nash, and established their importance as an object of study in M. Kontsevich' s motivic integration theory. Several results point out that jet schemes carry a rich amount of geometrical information about the original object they stem from, whereas, from an algebraic point of view, little is know about them. In this talk, after recalling some basic facts about classical determinantal varieties and jet schemes, we study some algebraic properties of jet schemes ideals of pfaffian varieties focusing on the problem of their irreducibility. This is a joint work with E. De Negri (Univ. Genova). Divendres 13 d'abril, 15h, Aula T2, FMI-UB Roser Homs Pons Universitat de Barcelona Computing Gorenstein colength of Artin rings In this talk, we will introduce the notion of Gorenstein colength of Artin local k-algebras to measure how far are such objects from being Gorenstein. Then we will see a characterization of k-algebras of low colengths (0, 1 and 2) in terms of their Macaulay's inverse systems. We provide effective algorithms to compute colength for low cases and discuss the problem for higher colengths. Finally, we will define the minimal Gorenstein cover variety. Divendres 27 d'abril, 15h, Aula T2, FMI-UB Eva Miranda Universitat Politècnica de Catalunya Desingularizing singular symplectic structures In joint work with Victor Guillemin and Jonathan Weitsman [GMW] we presented a desingularization technique for Poisson structures (including b-Poisson manifolds [GMP]) which are symplectic away from a smooth hypersurface and meet some transversality requirements. These "singular symplectic structures" appear modelling some problems in celestial mechanics (like the 3-body problem or the elliptic restricted 3-body problem) where the singularities are associated to the so-called line at infinity or collision set. The desingularization technique (deblogging) associates a family of symplectic structures to singular symplectic structures with even exponent (the so-called $$b^{2k}-$$symplectic structures) and a family of folded symplectic structures for odd exponent ($$b^{2k+1}-$$symplectic structures) and has good convergence properties. In this talk I will briefly present this desingularization technique and explore novel applications to equivariant singular symplectic geometry. Time permitting, I will also present new applications in the study of their Hamiltonian Dynamics (including the quest of periodic orbits) taking the restricted 3-body problem as guinea pig [AFKP]. [AFKP] P. Albers, U. Frauenfelder, O. van Koert, G. P. Paternain. Contact geometry of the restricted three-body problem. Comm. Pure App. Math. 65, (2012) [KMS] A. Kiesenhofer, E. Miranda, G. Scott. Action-angle variables and a KAM theorem for b-Poisson manifolds, J. Math. Pures Appl. (9) 105 (2016) [GMP] V. Guillemin, E. Miranda, A.R. Pires, Symplectic and Poisson geometry on b-manifolds, Adv. Math. 264 (2014). [GMW] V. Guillemin, E. Miranda, J. Weitsman, Desingularizing $$b^m$$-symplectic structures, IMRN. [GMW2] V. Guillemin, E. Miranda, J. Weitsman, On geometric quantization of b-symplectic manifolds, accepted for Adv. Math. Divendres 4 de maig, 15h, Aula T2, FMI-UB Guillem Blanco Universitat Politècnica de Catalunya TBA Divendres 11 de maig, 15h, Aula T2, FMI-UB Cédric Oms Universitat Politècnica de Catalunya TBA Divendres 25 de maig, 15h, Aula T2, FMI-UB Tomasz Szemberg Uniwersytet Pedagogiczny w Krakowie, Polònia TBA Divendres 1 de juny, 15h, Aula T2, FMI-UB Luca Schaffler University of Massachusetts Amherst, USA TBA Divendres 8 de juny, 15h, Aula T2, FMI-UB Xavier Gómez Mont CIMAT, Mèxic TBA Divendres 8 de juny, 16h, Aula T2, FMI-UB Marc Masdeu UAB TBA Divendres 15 de juny, 15h, Aula T2, FMI-UB

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