Seminari de Geometria Algebraica de Barcelona
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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 Roberta di Gennaro U. Napoli Federico II, Itàlia Contact: miro at ub.edu TBA Divendres 17 de gener, 15h00, Aula T2, FMI-UB

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