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 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 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 TBA Divendres 1 de desembre, 15h, Aula T2, FMI-UB Matthias Nickel Goethe Universität Frankfurt am Main TBA Divendres 15 de desembre, 15h, Aula T2, FMI-UB Workshop in Complex Algebraic Geometry 5 - 9 de febrer 2018, Aula B5, UB

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