Seminari de Geometria Algebraica de Barcelona
 UB UPC UAB
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 Seminari de Geometria Algebraica 2019/2020 Conferenciant Títol Data i hora Women in Geometry and Topology 25 a 27 de setembre, CRM Joaquim Roé UAB Newton-Okounkov polygons Since the formalization of Newton-Okounkouv bodies by Kaveh-Khovanski and Lazarsfeld-Mustata about ten years ago, we have learned a lot about their shape and how it reflects the properties of varieties and line bundles, espcially concerning cohomology and positivity. After Jow's proof that the set of all Newton-Okounkov bodies of a given line bundle is a complete numerical invariant (and local variants of this result) it is natural to ask about this set, for instance, how do bodies of a given line bundle vary by chosing different flags/valuations? And how can one extract information about the line bundle from its collection of bodies? I will look at some well known results on Newton-Okounkov bodies from this point of view, and report on work in progress with Moyano-Fernández, Nickel and Szemberg for the case of surfaces and Newton-Okounkov polygons. Divendres 11 d'octubre, 15h10, Aula T2, FMI-UB Irene Spelta Università di Pavia, Itàlia Contact: jcnaranjo at ub.edu POSPOSAT Divendres 18 d'octubre, 15h00, Aula T2, FMI-UB Olivier Martin U. Chicago - Collège de France Contact: jcnaranjo at ub.edu The degree of irrationality of most abelian surfaces is 4 The degree of irrationality of a complex projective $$n$$-dimensional variety $$X$$ is the minimal degree of a dominant rational map from $$X$$ to $$n$$-dimensional projective space. It is a birational invariant that measures how far $$X$$ is from being rational. Accordingly, one expects the computation of this invariant in general to be a difficult problem. Alzati and Pirola showed in 1993 that the degree of irrationality of any abelian $$g$$-fold is at least $$g+1$$ using inequalities on holomorphic length. Tokunaga and Yoshihara later proved that this bound in sharp for abelian surfaces and Yoshihara asked for examples of abelian surfaces with degree of irrationality at least $$4$$. Recently, Chen and Chen, Stapleton showed that the degree of irrationality of any abelian surface is at most $$4$$. In this work I provide the first examples of abelian surfaces with degree of irrationality $$4$$. In fact, I show that most abelian surfaces have degree of irrationality $$4$$. For instance, a very general $$(1,d)$$-polarized abelian surface has degree of irrationality $$4$$ if $$d$$ does not divide $$4$$. We will give the complete proof since it is very short and uses nothing beyond Mumford's theorem on rational equivalences of zero-cycles on surfaces with $$p_g>0$$. Divendres 25 d'octubre, 15h00, Aula T2, FMI-UB Simone Marchesi UB Terao's conjecture holds for RUA arrangements One of the most famous open problems in the topic of hyperplane arrangements is the so called Terao's conjecture, which states that the freedom of an arrangement depends on its combinatorics. After giving an introduction to the problem, we will focus our attention on the special case of triangular arrangements in the projective plane, i.e. with all the lines passing through three fixed points. We prove that the conjecture holds for a central family, calles Roots of Unity Arrangements, which also provides a counterexample to the weak conjecture. This is a joint work with Jean Vallès. Divendres 8 de novembre, 15h00, Aula T2, FMI-UB Irene Spelta Università di Pavia, Itàlia Contact: jcnaranjo at ub.edu Infinitely many totally geodesic subvarieties via Galois coverings of Elliptic curves We will speak about totally geodesic subvarieties of $$\mathcal{A}_{g}$$ which are generically contained in the Torelli locus. Coleman-Oort conjecture says that for genus $$g$$ large enough such varieties should not exist. Nevertheless if $$g\leq7$$ there are examples obtained as families of Jacobians of Galois coverings of curves $$f:C\to C'$$, where $$C'$$ is a smooth curve of genus $$g'$$. All of them satisfy a sufficient condition, which we will denote by ($$\ast$$). We will describe how strong is the condition ($$\ast$$). First, it gives us a bound on the genus $$g'$$ which we use to say that there are only 6 families of Galois coverings of curves of $$g'\geq1$$ which yield Special subvarieties of $$\mathcal{A}_{g}$$. Then we use ($$\ast$$) again to study the Prym maps of the families described above: we will prove that they are fibered, via their Prym map, in curves which are totally geodesic. In this way, we get infinitely many new examples of totally geodesic subvarieties of $$\mathcal{A}_{2},\ \mathcal{A}_{3}$$ and $$\mathcal{A}_{4}$$. Divendres 15 de novembre, 15h00, Aula T2, FMI-UB Simone Melchiorre Chiarello U. Geneva, Suïssa Contact: eva.miranda at upc.edu Quasi-symplectic reductions and equivariant cohomology Symplectic reductions provide an elegant framework for the study of moduli spaces. In particular, Atiyah and Bott in their famous paper "The Yang-Mills equations over Riemann surfaces" proved that the moduli space of flat unitary connections is a particular symplectic reduction, of an infinite-dimensional affine space via an infinite-dimensional group. This gives good information about the cohomology of such moduli spaces, thanks to Kirwan's surjectivity theorem. However, when we try to display the same moduli space via finite-dimensional manifolds, we fall into the domain of quasi-symplectic reductions. I will talk about the issues which arise in this setting, which are closely related to higher algebraic structures on equivariant cohomology. This exposes a work in progress with A. Alekseev. Divendres 22 de novembre, 15h00, Aula T2, FMI-UB Francesc Planas UPC Noetherian rings of low global dimension and syzygetic prime ideals In this talk we present some recent work on Noetherian rings of low global dimension Concretely, let $$R$$ be a Noetherian ring. We prove that $$R$$ has global dimension at most two if, and only if, every prime ideal of $$R$$ is of linear type. Similarly, one proves that $$R$$ has global dimension at most three if, and only if, every prime ideal of $$R$$ is syzygetic. As a consequence, one derives a characterization of these rings using the André-Quillen homology. The extension of this result to the non-Noetherian setting is an open question by now. Divendres 29 de novembre, 15h00, Aula T2, FMI-UB Xevi Guitart UB Endomorphism algebras of geometrically split abelian surfaces over $$\mathbb{Q}$$ An abelian surface defined over the rationals is said to be geometrically split if its base change to the complex numbers is isogenous to a product of elliptic curves. The aim of the talk is to explain the computation of all possible endomorphism algebras of geometrically split abelian surfaces defined over $$\mathbb{Q}$$. This is joint work with Francesc Fité. Divendres 10 de gener, 15h00, Aula T2, FMI-UB Roberta di Gennaro U. Napoli Federico II, Itàlia Contact: miro at ub.edu Line arrangements and Lefschetz properties In (Di Gennaro, Ilardi, Valles, 2014) and (Di Gennaro, Ilardi, 2018), the relationship between line arrangements such that the logarithmic bundle has not balanced splitting on a general line and suitable ideals of power of linear forms failing the Strong Lefschetz Property at range 2 is highlighted. Here, we explain these relation and generalize it to ideals failing SLP at any range $$k$$, by introducing the so-called generalized logarihtmic bundle. We are also interested in the relation with the unexpected curves (Cook, Harbourne, Migliore, Nagel, 2017) and in suitable examples. Divendres 17 de gener, 15h00, Aula T2, FMI-UB Kolja Knauer UB The Ehrhart polynomial of a lattice path matroid Lattice path matroids are very nice and manageable matroids, that form a special class of matroids. The base polytope of a matroid is the convex hull of the incidence vectors of its bases and the Ehrhart polynomial is a polynomial associated to any integer polytope. It counts the integer points in a certain way. In this talk we will discuss several open questions about the Ehrhart polynomial and their specialization to base polytopes of lattice path matroids. Many of our results rely on the distributive lattice structure carried by the bases of a lattice path matroid - a property shared by positroids. Joint with Leoardo Martinez Sandoval and Jorge Ramirez Alfonsin. Divendres 7 de febrer, 15h00, Aula T2, FMI-UB Ignacio García Marco U. de la Laguna, Tenerife. Contact: kolja.knauer at lis-lab.fr Lower Bounds by Birkhoff Interpolation This talk deals with the problem of obtaining lower bounds for the representation of a univariate polynomial $$f \in \mathbb{R}[x]$$ of degree $$d$$ under the form: $$f(x)=\sum_{i=1}^{\ell} \alpha_i (x+a_i)^{e_i},$$ where the $$\alpha_i, a_i$$ are real constants and the exponents $$e_i$$ nonnegative integers. More precisely, we give families of polynomials such that the number $$\ell$$ of terms required in such a representation must be at least of order $$d$$. This is clearly optimal up to a constant factor. Previous lower bounds for this problem were only of order $$\Omega(\sqrt{d})$$. Our lower bound results are specific to polynomials with real coefficients. It would be interesting to obtain similar lower bounds for other fields, e.g., finite fields or the field of complex numbers. In order to obtain our results we relate our problem with one of Birkhoff interpolation and then we apply some well-known results in real Birkhoff interpolation. Finally, we will discuss the relation of this problem with an algebraic version of the $$\mathcal{P}$$ vs. $$\mathcal{NP}$$ problem. This is a joint work with Pascal Koiran and Timothée Pecatte. Divendres 14 de febrer, 15h00, Aula T2, FMI-UB András Lőrincz Max Planck Institute for Mathematics in the Sciences, Leipzig. Contact: josep.alvarez at upc.es Local cohomology modules on a class of representations Let $$X$$ be an irreducible representation of a reductive group $$G$$ that has finitely many orbits. In this talk I will discuss results on local cohomology modules on $$X$$ supported in orbit closures, by describing their explicit $$D$$-module and $$G$$-module structures. The approach is through the study of the category of $$G$$-equivariant $$D$$-modules on $$X$$, which has a quiver description. Other applications include Lyubeznik numbers and intersection cohomology groups of orbit closures. Divendres 21 de febrer, 15h00, Aula T2, FMI-UB Ana Peón Nieto UPC Equality of the wobbly and shaky loci The geometric Langlands correspondence (GLC) generalises the fact that a rank one local system on a smooth projective curve uniquely extends to its Jacobian. According to the GLC, local systems of rank $$n$$ should produce $$D$$-modules on the moduli space of $$\operatorname{GL}(n,\mathbb{C})$$-bundles. Donagi-Pantev devised a programme aiming at deducing the GLC from the rank one case using the non abelian Hodge correspondence and Higgs bundles. This requires, as a first step, to understand the resolution of a rational map. In this talk I will explain a joint result with Christian Pauly, and the applicability of the ideas therein towards the proof of a conjecture by Donagi and Pantev, according to which the indeterminacy locus of the aforementioned rational map can be described in terms of wobbly bundles (namely, bundles with non zero nilpotent Higgs fields). Divendres 28 de febrer, 15h00, Aula T2, FMI-UB Dan Agüero IMPA, Brasil. Contact: rubio at mat.uab.cat TBA Divendres 6 de març, 15h00, Aula T2, FMI-UB Makoto Enokizono Tokyo U. of Science, Japó. Contact: miguel.angel.barja at upc.edu Slope inequality for fibered surfaces and Durfee's conjecture for surface singularities In 1978, A. H. Durfee conjectured that the signature of the Milnor fiber of a smoothing of a normal surface singularity is always negative. In this talk, I will explain that this conjecture holds true for two-dimensional isolated complete intersection singularities as an application of the slope inequality for certain fibered surfaces. The first half of the talk is dedicated to the introduction of the slope inequality of fibered surfaces and Durfee's conjecture. In the second half of the talk, I will explain the strategy of the proof of the main theorem and some conjecture (higher dimensional analogue of the slope inequality, Durfee's conjecture and so on). Divendres 13 de març, 15h00, Aula T2, FMI-UB Maxim Braverman Northeastern U., EEUU Contact: eva.miranda at upc.edu Geometric quantization of non-compact and b-symplectic manifolds We introduce a method of geometric quantization for of non-compact symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (APS) boundary value problem. We then apply it to a class of compact manifolds with singular symplectic structure, called b-symplectic manifolds. We show further that b-symplectic manifolds have canonical Spin-c structures in the usual sense, and that the APS index above coincides with the index of the Spin-c Dirac operator. We show that if the manifold is endowed with a Hamiltonian action of a compact connected Lie group with non-zero modular weights, then this method satisfies the Guillemin-Sternberg "quantization commutes with reduction" property. In particular our quantization coincides with the formal quantization defined by Guillemin, Miranda and Weitsman, providing a positive answer to a question posed in their paper. Divendres 20 de març, 15h00, Aula T2, FMI-UB Hanine Awada, Montpeller. Benoît Cadorel, Nancy. Carlos D'Andrea, Barcelona. Andreas Hoering, Niça. Henri Guenancia, Toulouse. Jorge Pereira, IMPA. SMGA 6 25-26 de març, Montpeller. Carolina Benedetti U. de los Andes, Bogotá, Colombia. Contact: kolja.knauer at lis-lab.fr TBA Divendres 27 de març, 15h00, Aula T2, FMI-UB Mircea Mustață U. Michigan, EEUU. Contact: josep.alvarez at upc.edu TBA Divendres 17 d'abril, 12h00, Col·loqui FME. Tong Zhang East China Normal U. Contact: miguel.angel.barja at upc.edu TBA Divendres 24 d'abril, 15h00, Aula T2, FMI-UB Nuno Romão U. Göttingen, Alemanya. Contact: eva.miranda at upc.edu TBA Divendres 8 de maig, 15h00, Aula T2, FMI-UB Lawrence Ein U. Illinois at Chicago, EEUU. Contact: marta.casanellas at upc.edu TBA Divendres 22 de maig, 15h00, Aula T2, FMI-UB

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