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  • On geometric quantisation of integrable systems with singularities

     Barbieri Solha, Romero
    Defense's date: 2013-10-21
    Universitat Politècnica de Catalunya
    Theses

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  • Hodge numbers of irregular varieties and fibrations  Open access

     Gonzalez Alonso, Victor
    Defense's date: 2013-07-08
    Universitat Politècnica de Catalunya
    Theses

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    In this thesis we study the geography of irregular complex projective (or compact Kähler) varieties, paying special attention to the existence of fibrations. The thesis is divided into two parts. In the first one we consider irregular varieties of arbitrary dimension, looking for bounds for the Hodge numbers in the absence of fibrations. In first place, by truncating the BGG complex of the variety (an object recently introduced by Lazarsfeld and Popa), we get lower bounds on the partial Euler characteristics. In order to improve these first results, we define the higher-rank derivative complexes (generalizing the derivative complex introduced by Green and Lazarsfeld). We study their exactness by means of the Eagon-Northcott complexes, and we obtain some inequalities between the Hodge numbers of varieties admitting some kind of subspaces of 1-forms (¿non-degenerate subspaces¿). In the case of subvarieties of Abelian varieties, the existence of non-degenerate subspaces of any dimension allows us to obtain better inequalities than in the general case. In the case of h^(2,0), a different method gives a much better result. In fact, the bound is much stronger, and the only hypothesis needed is the non-existence of higher irrational pencils (a priori, less restrictive than the existence of non-degenerate subspaces). To close, using the Grassmannian BGG complex (a generalization of the BGG complex that aggregates all the higher-rank derivative complexes) and computing the Chern classes of its last cokernel, we recover the same bound for h^(2,0) using the general results mentioned in the previous paragraph. In the second part, the scope is restricted to surfaces fibred over a curve. We look for upper bounds for the relative irregularity in terms of properties of the general fibre, in the spirit of the inequality obtained by Xiao for non-isotrivial fibrations over a rational curve. Xiao conjectured the same inequality to hold for fibrations over any base, but Pirola found a counterexample. After that, a corrected conjecture was proposed. The result obtained in this thesis is a bound depending on the genus and the Clifford index of a general fibre, which coincides with the corrected conjecture in the case of maximal Clifford index. We have used several techniques in our proof. On the one hand, the ¿adjoint images¿ play a crucial role. The adjoint images were introduced by Collino and Pirola to study infinitesimal deformations of smooth curves, and generalized later by Pirola and Zucconi to higher-dimensional varieties. In this thesis we construct the ¿global adjoint map¿, which allows to find subspaces with vanishing adjoint image assuming that the kernel of the infinitesimal deformation has dimension (at least) half the genus of the cruve. More generally, the global adjoint map can also be defined for infinitesimal deformations of irregular varieties of any dimension, and allows to find numerical conditions that guarantee the existence of subspaces with vanishing adjoint. On the other hand, we have extended to arbitrary (one-dimensional) families of curves some well-stablished concepts of infinitesimal deformations, related with the bicanonical embedding of the curve. As the global adjoint map, some of these constructions can also be extended to families of irregular varieties of arbitrary dimension. Finally, all these previous constructions lead to a structural result for fibrations supported on a relatively rigid divisor. With this result we can treat some cases of the conjecture of Xiao. The remaining cases are solved using an inequality for the rank of an infinitesimal deformation in terms of a supporting divisor (its degree and the dimension of its complete linear series). This inequality, which we reprove, is originally due to Ginensky.

  • GEOMATRIA ALGEBRAICA, SIMPLECTICA, ARITMETICA Y APLICACIONES

     Alberich Carramiñana, Maria; Amoros Torrent, Jaume; Barja Yañez, Miguel Angel; Elgueta Montó, Josep; Fernández Sánchez, Jesús; Ventura González Alonso, Daniel; Barbieri Solha, Romero; Miranda Galceran, Eva; Pascual Gainza, Pedro; Roig Marti, Agustin; Roig Maranges, Abdo; Feliu Trijueque, Elsienda; Gálvez Carrillo, Maria Immaculada; Casanellas Rius, Marta
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  • Some remarks on Cartan-Eilenberg categories

     Pascual Gainza, Pedro
    Collectanea mathematica
    Date of publication: 2012-05
    Journal article

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    In this note we collect some remarks and examples on Cartan–Eilenberg categories.

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    On cdh-descent  Open access

     Pascual Gainza, Pedro
    Date: 2011-09
    Report

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    These are the expanded notes of some talks in the working seminar group on Algebraic Geometry at the Universitat Polit ecnica de Catalunya in 2008. The aim was to understand Haesemeyer's descent arugument, so we present the main results of Haesemeyer and his collaborators on cdh-descent following very closely their papers

  • A Cartan-Eilenberg approach to homotopical algebra

     Guillén Santos, Francisco; Navarro Aznar, Vicente; Pascual Gainza, Pedro; Roig Marti, Agustin
    Journal of pure and applied algebra
    Date of publication: 2010
    Journal article

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    In this paper we propose an approach to homotopical algebra w here the basic ingredient is a category with two classes of distinguished morphisms: s trong and weak equivalences. These data determine the cofibrant objects by an extension property ana logous to the classical lifting property of projective modules. We define a Cartan-Eilenberg categor y as a category with strong and weak equivalences such that there is an equivalence of categorie s between its localisation with respect to weak equivalences and the relative localisation of the subc ategory of cofibrant objets with respect to strong equivalences. This equivalence of categories allow s us to extend the classical theory of derived additive functors to this non additive setting. The main exa mples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latt er case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theor em

  • Theta-duality in abelian varieties and the bicanonical map of irregular varieties  Open access  awarded activity

     Lahoz Vilalta, Marti
    Defense's date: 2010-05-18
    Department of Applied Mathematics I, Universitat Politècnica de Catalunya
    Theses

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    The first goal of this Thesis is to contribute to the study of principally polarized abelian varieties (ppav), especially to the Schottky and the Torelli problems. Ppav admit a duality theory analogous to that of projective spaces, where the role played by hyperplanes in projective spaces is played by divisors representing the principal polarization. Thus, given a subvariety Y of a ppav, we can define its thetadual T(Y) as the set of divisors representing the principal polarization that contain this subvariety. This set admits a natural schematic structure (as defined by Pareschi and Popa). Jacobian and Prym varieties are classical examples of ppav constructed from curves. Besides, they are interesting because some properties of the curves involved in their construction are reflected in their geometry or in the geometry of some special subvarieties. For example, in the case of Jacobians we have the BrillNoether loci Wd ( W1 corresponds to the AbelJacobi curve) and in the case of Pryms we have the AbelPrym curve C. In chapter III, we study the schematic structure of the thetadual of the BrillNoether loci Wd and the AbelPrym curve. In the first case, we obtain with different methods, the result of Pareschi and Popa T(Wd)= Wgd1. In the case of the AbelPrym curve C, we get that T(C)=V², where V² is the second PrymBrillNoether locus with the schematic structure defined by Welters. Pareschi and Popa have proved a result for ppavs analogous to the Castelnuovo Lemma for projective spaces. That is, if (A,Θ) is a ppav of dimension g, then g+2 distinct points in general position with respect to Θ, but in special position with respect to 2Θ, have to be contained in a curve of minimal degree in A, i.e. an AbelJacobi curve. In particular, they obtain a Schottky result because A has to be a Jacobian variety and a Torelli result, because the curve is the intersection of all the divisors in |2Θ| that contain the g+2 points. In chapter IV, as Eisenbud and Harris have done in the projective Castelnuovo Lemma, we extend this result to possibly nonreduced finite schemes. The second goal of this thesis is the study of varieties of general type. Almost by definition, pluricanonical maps are the essential tool to study them. One of the main problems in this area is to find geometric or numerical conditions to guarantee that the mth pluricanonical map (for low m) induces a birational equivalence with its image. The classification of surfaces whose bicanonical map is nonbirational has attracted considerable interest among algebraic geometers. In chapter V, we give a sufficient numerical condition for the birationality of the bicanonical map of irregular varieties of arbitrary dimension. We also prove that, if X is a primitive variety, then it only admits very special fibrations to other irregular varieties. For primitive varieties we get that the following are equivalent: X is birational to a divisor Θ in an indecomposable ppav, the irregularity q(X) > dim X and the bicanonical map is nonbirational. When X is a primitive variety of general type and q(X) = dim X we prove, under certain conditions over the Stein factorization of the Albanese map, that the only possibility for the bicanonical map being nonbirational is that X is a double cover branched along a divisor in |2Θ|. These results extend to arbitrary dimension, wellknown theorems in the case of surfaces and curves.

    El primer objectiu d'aquesta tesi és contribuir a l'estudi de les varietats abelianes principalment polaritzades (vapp), especialment als problemes de Schottky i Torelli. Les vapp admeten una teoria de dualitat anàloga a la dualitat dels espais projectius, on el paper que juguen els hiperplans de l'espai projectiu és substituït pels divisors que representen la polarització principal. Així doncs, donada una subvarietat Y d'una vapp, podem definir el seu thetadual T(Y) com el conjunt dels divisors que representen la polarització principal i contenen aquesta subvarietat. Aquest conjunt admet una estructura esquemàtica natural (tal i com la defineixen Pareschi i Popa). Les varietats Jacobianes i de Prym són exemples clàssics de vapp construïdes a partir de corbes. A més, són interessants perquè certes propietats de les corbes involucrades es veuen reflectides en elles o en algunes subvarietats especials. Per exemple, en el cas de les Jacobianes tenim els llocs de BrillNoether Wd ( W1 correspon a la corba d'AbelJacobi) i en el cas de les Pryms tenim la corba d'AbelPrym C. Al capítol III de la tesi s'estudia l'estructura esquemàtica del thetadual dels llocs de BrillNoether Wd i de la corba d'AbelPrym. En el primer cas, es reobté amb uns altres mètodes, el resultat de Pareschi i Popa T(Wd)= Wgd1. En el cas de la corba d'AbelPrym C, s'obté que T(C)=V², onV² és el segon lloc de PrymBrillNoether amb l'estructura esquemàtica definida per Welters. Pareschi i Popa han demostrat un resultat anàleg per les vapp al Lemma de Castelnuovo pels espais projectius. És a dir, si (A,Θ) és una vapp de dimensió g, aleshores g+2 punts en posició general respecte Θ, però en posició especial respecte 2Θ, han d'estar continguts en una corba de grau minimal a A, i.e. una corba d'AbelJacobi. En particular, s'obté un resultat de Schottky ja que A ha de ser una Jacobiana i un resultat de Torelli, ja que la corba és la intersecció de tots els divisors de |2Θ| que contenen els g+2 punts. Al capítol IV, tal i com Eisenbud i Harris van fer en el cas projectiu, s'estén aquest resultat a esquemes finits possiblement no reduïts. El segon objectiu d'aquesta tesi és contribuir a l'estudi de les varietats de tipus general. Pràcticament per definició, les aplicacions pluricanòniques són essencials pel seu estudi. Un dels problemes principals de l'àrea és donar condicions geomètriques o numèriques per assegurar que la mèsima aplicació pluricanònica (per m baix) indueix una equivalència biracional amb la imatge. La classificació de les superfícies que tenen l'aplicació bicanònica no biracional ha atret l'atenció de molts geòmetres algebraics. Al capítol V, es dóna un criteri numèric suficient per assegurar la biracionalitat de l'aplicació bicanònica de les varietats irregulars de dimensió arbitrària. També es demostra que si X és una varietat primitiva, aleshores només admet fibracions molt especials a altres varietats irregulars. Per aquestes varietats s'obté que és equivalent que X sigui biracional a un divisor Θ en una vapp indescomponible, a què la irregularitat q(X) > dim X i l'aplicació bicanònica sigui no biracional. Quan X és una varietat primitiva de tipus general i q(X) = dim X es demostra sota certes condicions de la descomposició de Stein del morfisme d'Albanese, que l'única possibilitat per tal que l'aplicació bicanònica sigui no biracional és que X sigui un recobriment doble sobre una vapp ramificat al llarg d'un divisor a |2Θ|. Aquest resultats estenen a dimensió arbitrària, teoremes ben coneguts en el cas de superfícies i corbes.

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    Report of the Seminar on Motivic Cohomology  Open access

     Pascual Gainza, Pedro; del Ángel Rodríguez, Pedro Luis
    Date: 2010
    Report

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    A seminar on Motivic Cohomology took place at the Centro de Investigaci´on en Matem´aticas (CIMAT), at Guanajuato (M´exico) during the month of August 2010. The aim of the seminar was to cover the definition and main properties of motivic cohomology of smooth schemes over a perfect field as defined by V. Voevodsky, and its relation to ´etale cohomology, Bloch’s higher Chow groups and Milnor’s algebraic K-theory, based on the book [MVW]. The seminar was addressed to an audience with a basic knowledge of algebraic geometry and classical motives. It was not possible to cover all proofs of the different results presented along the sessions of the seminar, but we think that it emerged a clear picture of the subject. We thank all participants by their enthusiasm and specially Abd´o Roig, who has read a first draft of this report and contributed with his remarks. The seminar began on August 2 and ended on September 1. It consisted of ten sessions of one hour and a half according to the following programme and speakers.

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    Review of the book: "Category Theory"  Open access

     Pascual Gainza, Pedro
    Date: 2010
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    Book Review Category Theory, Second Edition Steve Awodey Oxford Logic Studies 52. Oxford University Press, 2010. ISBN 978 0 19 958736 0

  • ESTRUCTURAS NO ABELIANAS EN GEOMETRIA Y TOPOLOGÍA ALGEBRAICAS

     Roig Marti, Agustin; Roig Maranges, Abdo; Pascual Gainza, Pedro
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    Some remarks on Cartan-Eilenberg categories  Open access

     Pascual Gainza, Pedro
    Date: 2010
    Report

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    In this note we collect some remarks and examples on Cartan-Eilenberg categories

  • The differentiable chain functor is not homotopy equivalent to the continuous chain functor

     Guillén, F; Navarro, V; Pascual Gainza, Pedro; Roig Marti, Agustin
    Topology and its applications
    Date of publication: 2009-01
    Journal article

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  • Emmy Noether i l'algebraització de la topologia

     Pascual Gainza, Pedro
    Jornada Noether
    Presentation's date: 2009-02-18
    Presentation of work at congresses

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    Suites spectrales  Open access

     Pascual Gainza, Pedro; Hamraoui, Hinda
    Date: 2009-09
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    Ces notes sont une version élargie des exposés soutenus par les auteurs dans divers séminaires sur la K-théorie algébrique développés à Barcelone et Casablanca les années 2007 et 2008. On suit de très proche l'article de J. Boardaman, [B], pour la théorie générale, et celui de R. Thomason, [T], pour les appplications à la cohomologie généralisée de faisceaux

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    Cartan-Eilenberg categories and descent categories  Open access

     Pascual Gainza, Pedro
    Date: 2009-09
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    Contents 1. Introduction 2. Remarks on localization of categories 3. Co¯brant objects 4. Cartan-Eilnberg categories 5. Minimal models: Sullivan categories 6. Models of functors 7. Cotriples and Cartan-Eilenberg categories

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  • Sur les catégories triangulées

     Pascual Gainza, Pedro
    Université Hassan II
    Presentation's date: 2009-02-11
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  • Acyclic models, monoidal functors and chain operads

     Guillén, F Santos; Navarro, V; Pascual Gainza, Pedro; Roig Marti, Agustin
    Canadian journal of mathematics. Journal canadien de mathématiques
    Date of publication: 2008-01
    Journal article

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  • Differential Galois Theory and Lie-Vessiot Systems  awarded activity

     Blazquez Sanz, David
    Defense's date: 2008-07-07
    Department of Applied Mathematics II, Universitat Politècnica de Catalunya
    Theses

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  • Categories de descens: aplicacions a la teoria K algebraica

     Rubio Pons, Llorenç
    Defense's date: 2008-07-08
    School of Mathematics and Statistics (FME), Universitat Politècnica de Catalunya
    Theses

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  • A Cartan-Eilenberg Approach to Homotopical Algebra

     Francisco, Guillén; Navarro, V; Pascual Gainza, Pedro; Roig Marti, Agustin
    Date: 2008-10
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  • The differentiable chain functor is not homotopy equivalent to the continuous chain functor

     Guillén, F; Navarro, V; Pascual Gainza, Pedro; Roig Marti, Agustin
    Date: 2008-09
    Report

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  • The differentiable chain functor is not homotopy equivalent to the continuous chain functor

     Guillén, F.; Navarro, V; Pascual Gainza, Pedro; Roig Marti, Agustin
    Date: 2008-09
    Report

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  • Descente cohomologieu ET K-théorie

     Pascual Gainza, Pedro
    Université Hassan II
    Presentation's date: 2008-08-01
    Presentation of work at congresses

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  • Modèles acycliques et catégories de cartan-Eilenberg

     Pascual Gainza, Pedro
    Université Hassan II
    Presentation's date: 2008-08-01
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  • Cartan-Eilenberg categories

     Pascual Gainza, Pedro
    Workshop on Derived Catgeories
    Presentation's date: 2007-07-01
    Presentation of work at congresses

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  • Las contribuciones de Gauss a la Física: Un panorama

     Pascual Gainza, Pedro
    Date of publication: 2007-08
    Book chapter

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  • Cartan-Eilenberg categories

     Pascual Gainza, Pedro; Guillén, F; Navarro, V; Roig Marti, Agustin
    Date: 2007-11
    Report

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  • The Catalan mathematician Ferran Sunyer i Balaguer (1912-1967) and the Prize in his honor

     Pascual Gainza, Pedro
    SCM/Notícies
    Date of publication: 2006-08
    Journal article

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  • Motivos mixtos y operads

     Guillén Santos, Francisco; Roig Marti, Agustin; Pascual Gainza, Pedro
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  • A Cartan-Eilenberg Approach to Homotopical Algebra

     Guillén, F; Navarro, V; Pascual Gainza, Pedro; Roig Marti, Agustin
    Date: 2006-07
    Report

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  • Moduli spaces and formal operads

     Guillén, Santos F; Navarro, V; Pascual Gainza, Pedro; Roig Marti, Agustin
    Duke mathematical journal
    Date of publication: 2005-08
    Journal article

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  • Algebraic K-Theory and cubical descent

     Pascual Gainza, Pedro; Llorenc, Rubió; Rubio Pons, Llorenç
    Date: 2005-06
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  • Geometria de superfícies. Una aproximació a la figura de Gauss

     Pascual Gainza, Pedro
    Butlletí de la Societat Catalana de Matemàtiques
    Date of publication: 2005-12
    Journal article

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  • Topologia

     Pascual Gainza, Pedro; Roig Marti, Agustin
    Date of publication: 2004-03
    Book

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  • Formality of Modular Operads

     Pascual Gainza, Pedro
    III Iberoamerican Congreso in Geometry
    Presentation's date: 2004-01-01
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  • Geometría y topología de las variedades algebraicas

     Barja Yañez, Miguel Angel; Roig Marti, Agustin; Pascual Gainza, Pedro; Elgueta Montó, Josep; Abío Roig, Ignasi; Amoros Torrent, Jaume; Casanellas Rius, Marta; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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  • Càlcul integral per a enginyers

     Pascual Gainza, Pedro; Bonet Reves, Carles; Compta Creus, Albert; Consul Porras, M. Nieves; Olle Torner, Maria Mercedes; Roig Marti, Agustin
    Date of publication: 2002-09
    Book

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  • Movilidad del profesorado en Programas de Doctorado

     Pascual Gainza, Pedro
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  • Fibraciones de Variedades Algebraicas y moduli

     Burgos Gil, Ignacio; Roig Marti, Agustin; Amoros Torrent, Jaume; Barja Yañez, Miguel Angel; Pascual Gainza, Pedro; Plans Berenguer, Bernat; Lahoz Vilalta, Marti; Mundet Riera, Ignasi; Alberich Carramiñana, Maria; Alvarez Montaner, Josep; Rubio Pons, Llorenç
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  • Topologia Algebraica

     Navarro, V; Pascual Gainza, Pedro
    Date of publication: 1999-07
    Book

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  • On motives and moduli spaces of stable vector bundles over a curve

     del Baño Rollin, Jose Sebastian
    Defense's date: 1998-06-25
    Department of Applied Mathematics I, Universitat Politècnica de Catalunya
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  • Fibraciones de variedades algebraicas

     Barja Yañez, Miguel Angel; Navarro Aznar, Vicente; Pascual Gainza, Pedro; Amoros Torrent, Jaume
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  • Cohomologia equivariant

     Pascual Gainza, Pedro
    Date: 1997-03
    Report

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  • Sobre la definición de K-Teoría Algebraica

     Pascual Gainza, Pedro
    Date: 1996-11
    Report

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