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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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  • Generalized Clifford-Severi inequality

     Barja Yañez, Miguel Angel
    Workshop on Algebraic Surfaces
    Presentation's date: 2012-03-30
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  • On the bicanonical map of irregular varieties

     Barja Yañez, Miguel Angel; Lahoz Vilalta, Marti; Naranjo del Val, Joan Carles; Pareschi, Giusseppe
    Journal of algebraic geometry
    Date of publication: 2012-07
    Journal article

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  • Premi extraordinari de Doctorat UPC

     Lahoz Vilalta, Martí; Barja Yañez, Miguel Angel
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  • Stability conditions and positivity of invariants of fibrations

     Barja Yañez, Miguel Angel; Stoppino, Lidia
    Date: 2012-12-19
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  • On the slope of higher dimensional fibrations

     Barja Yañez, Miguel Angel
    Workshop on Classical Algebraic Geometry
    Presentation's date: 2011-02-10
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  • Desplegament de l'exposició RSME imaginari a Catalunya. Fase 2011

     Xambó Descamps, Sebastian; Plans Berenguer, Bernat; Borras Sol, Julia; Barja Yañez, Miguel Angel; Thomas Arroyo, Federico; Quer Bosor, Jordi; Torras, Carme; Alberich Carramiñana, Maria
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  • 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.

  • DESPLEGAMENT DE L'EXPOSICIÓ RSME-IMAGINARY A CATALUNYA - FASE 2010

     Xambó Descamps, Sebastian; Torras, Carme; Thomas Arroyo, Federico; Barja Yañez, Miguel Angel; Plans Berenguer, Bernat; Quer Bosor, Jordi; Borras Sol, Julia; Alberich Carramiñana, Maria
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    On the bicanonical map of irregular varieties  Open access

     Barja Yañez, Miguel Angel; Lahoz Vilalta, Marti; Naranjo del Val, Joan Carles; Pareschi, Giusseppe
    Date: 2010-02
    Report

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    From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus 2: varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform

  • GEOMETRÍA DE VARIEDADES ALGEBRAICAS Y APLICACIONES

     Alberich Carramiñana, Maria; Barja Yañez, Miguel Angel; Elgueta Montó, Josep; Amoros Torrent, Jaume; Fernández Sánchez, Jesús; Gonzalez Alonso, Victor; Berna Sepulveda, Isabel Silvana; Ferragut Amengual, Antoni Manel; Casanellas Rius, Marta
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  • Slopes of trigonal fibred surfaces and of higher dimensional fibrations

     Barja Yañez, Miguel Angel; Stoppino, Lidia
    Annali della Scuola Normale Superiore di Pisa. Classe di scienze
    Date of publication: 2009
    Journal article

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  • GEOMETRIA DE VARIETATS I APLICACIONS

     Berna Sepulveda, Isabel Silvana; Barja Yañez, Miguel Angel; Casanellas Rius, Marta; Plans Berenguer, Bernat; Alvarez Montaner, Josep; Gonzalez Alonso, Victor; Elgueta Montó, Josep; Abío Roig, Ignasi; Amoros Torrent, Jaume; Fernández Sánchez, Jesús; Alberich Carramiñana, Maria
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  • Linear stability of projected canonical curves with applications to the slope of fibred surfaces

     Barja Yañez, Miguel Angel; Stoppino, L
    Journal of the Mathematical Society of Japan
    Date of publication: 2008-01
    Journal article

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  • Slope os trigonal fibred surfaces and of higher dimensional fibrations

     Barja Yañez, Miguel Angel; Stoppino, Lidia
    Date: 2008-07
    Report

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  • Higher dimensional slopes and a conjecture of Harris-Stankova-Frenkel

     Barja Yañez, Miguel Angel
    Workshop on algebraic Surfaces
    Presentation's date: 2008-10-30
    Presentation of work at congresses

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  • On the topological index of irregular surfaces

     Barja Yañez, Miguel Angel; Naranjo, J C; Pirola, G P
    Journal of algebraic geometry
    Date of publication: 2007-01
    Journal article

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  • Birational Local Torelli Theorem for 1+n forms

     Barja Yañez, Miguel Angel
    Oberwolfach workshop on Complex Algebraic Varieties
    Presentation's date: 2007-10-03
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  • Geometria i topologia de les varietats algebraiques

     Amoros Torrent, Jaume; Roig Marti, Agustin; Barja Yañez, Miguel Angel; Elgueta Montó, Josep; Abío Roig, Ignasi; Casanellas Rius, Marta; Alberich Carramiñana, Maria
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  • Geometría y topología de las variedades algebraicas

     Barja Yañez, Miguel Angel; Elgueta Montó, Josep; Alberich Carramiñana, Maria; Roig Marti, Agustin; Abío Roig, Ignasi; Amoros Torrent, Jaume; Casanellas Rius, Marta
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  • Geometría y Topología de Variedades Algebraicas y Simplécticas

     Casanellas Rius, Marta; Barja Yañez, Miguel Angel; Fernández Sánchez, Jesús
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  • Projecte BFM2003-06001

     Barja Yañez, Miguel Angel
    Date: 2006-03
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  • On the topological index of irregular surfaces

     Barja Yañez, Miguel Angel; Naranjo, J C; Pirola, G P
    Date: 2006-03
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  • The topological index of irregular surfaces

     Barja Yañez, Miguel Angel
    Workshop on complex algebraic varieties
    Presentation's date: 2006-05-05
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  • On the slope of nonÁlbanese fibrations

     Barja Yañez, Miguel Angel; Stoppino, L
    Date: 2005-06
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  • Review de l'article: Inequalities of Neother type for 3-folds of general type. Author: Chen, Meng

     Barja Yañez, Miguel Angel
    Mathematical reviews
    Date of publication: 2004-02
    Journal article

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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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  • Mathematische Zeitschrift

     Barja Yañez, Miguel Angel
    Collaboration in journals

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  • Annali della Scuola Normale Superiore di Pisa. Classe di scienze

     Barja Yañez, Miguel Angel
    Collaboration in journals

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  • Mathematische Zeitschrift

     Barja Yañez, Miguel Angel
    Collaboration in journals

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  • The representation problem of pairs of symmetric second order tensors in the context of solid mechanics

     Barja Yañez, Miguel Angel; Carol Vilarasau, Ignacio; Planas Vilanova, Francesc d'Assis; Rizzi, E
    Date: 2004-01
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  • Similarity pais of symmetric tensors in the context of Solid mechanics and further interpretations

     Barja Yañez, Miguel Angel; Planas Vilanova, Francesc d'Assis
    Date: 2003-07
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  • Regular canonical covers

     Barja Yañez, Miguel Angel
    Date: 2003-06
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  • Projective bundles of singular plane cubics

     Barja Yañez, Miguel Angel
    Date: 2003-09
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  • An algebraic proof of Litaka's conjecture C2.1

     Barja Yañez, Miguel Angel
    Date: 2003-09
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  • The representation problem of pairs of symmetric second-order tensors in the context of solid mechanics

     Barja Yañez, Miguel Angel; Carol Vilarasau, Ignacio; Planas Vilanova, Francesc d'Assis; Rizzi, E
    Date: 2003-12
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  • The representation problem of pairs of symmetric second-order tensors in the context of Solid Mechanics

     Barja Yañez, Miguel Angel; Carol Vilarasau, Ignacio; Planas Vilanova, Francesc d'Assis; Rizzi, E
    Date: 2003-07
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  • Àlgebra Lineal. Problemes resolts i comentats

     Barja Yañez, Miguel Angel; Garcia Planas, Maria Isabel; Hernando Martin, Maria Del Carmen; Magret Planas, Maria Dels Dolors; Planas Vilanova, Francesc d'Assis; Puig Pla, Carles Maria
    Date of publication: 2002-01
    Book

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  • The structure and slope of bielliptic fibrations

     Barja Yañez, Miguel Angel
    Proceedings of the American Mathematical Society
    Date of publication: 2001-12
    Journal article

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  • On the slope of fibred surfaces

     Barja Yañez, Miguel Angel; Zucconi, F
    Nagoya mathematical journal
    Date of publication: 2001-01
    Journal article

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  • Mathematical reviews

     Barja Yañez, Miguel Angel
    Collaboration in journals

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  • Mathematical reviews

     Barja Yañez, Miguel Angel
    Collaboration in journals

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