Le groupe de Cremona de rang 2 est le groupe des transformations birationnelles du plan projectif. Le but de cette thèse est d'étudier et de construire des espaces hyperboliques sur lesquels le groupe de Cremona agit et qui permettent de mettre en œuvre des méthodes provenant de la théorie géométrique des groupes. Il est connu depuis une dizaine d'année que le groupe de Cremona agit sur un espace hyperbolique H analogue au plan hyperbolique classique mais de dimension infinie. Dans un premier temps, nous montrons que le groupe de Cremona défini sur un corps quelconque n'est pas simple en le faisant agir sur cet espace hyperbolique. Ceci prolonge un résultat déjà connu dans le cas d'un corps de base algébriquement clos. Nous nous intéressons ensuite à un graphe construit par D. Wright sur lequel agit le groupe de Cremona. Nous montrons qu'il ne possède pas la propriété que nous souhaitions, à savoir qu'il n'est pas hyperbolique au sens de Gromov. Nous construisons également un domaine fondamental pour l'action du groupe de Cremona sur H via la méthode des cellules de Voronoï. Nous caractérisons les applications du groupe de Cremona qui correspondent à un domaine adjacent au domaine fondamental. Cela nous permet de prouver que le graphe de Wright est quasi-isométrique au graphe dual à ce pavage. Nous obtenons ainsi une manière de retrouver le graphe de Wright dans H. Nous montrons enfin qu'en modifiant ce graphe dual, nous obtenons un graphe hyperbolique au sens de Gromov. Dans une dernière partie, nous nous intéressons à une autre propriété naturelle qui est la propriété CAT(0). Nous construisons un complexe cubique CAT(0) de dimension infinie muni d'une action naturelle du groupe de Cremona. / The Cremona group of rank 2 is the group of birational transformations of the projective plane. The aim of this thesis is to study and build some hyperbolic spaces with a natural action of the Cremona group. We want these spaces to have good geometric properties in order to use methods coming from geometric group theory. It is known that the Cremona group acts on a hyperbolic space H which is similiar to the classical hyperbolic plane but in infinite dimension. First, using this action, we show that the Cremona group is not simple over any field. This extends previous results over an algrebraic closed field. Then we study the Wrigth's graph. We show that it doesn't have the property we are looking for, in the sense that it is not Gromov hyperbolic. We build a fundamental domain for the action of the Cremona group on H 8 via Voronoï's cells. We characterize birational tranformations that correspond to adjacent domains of the fundamental domain. This allows us to prove that the Wright's graph is quasi-isometric to the dual graph of this tessellation. It's give us a way of realizing the Wright's graph inside H. Finally, we show that by modifying the dual graph we obtain a Gromov hyperbolic graph. In the last part, we are interested in another classical property which is the CAT(0) property. We build an infinite dimensional CAT(0) cubical complex which comes with a natural action of the Cremona group.

Le groupe de Cremona en n variables Cr_n(C) est le groupe des transformations birationnelles de l'espace projectif complexe de dimension n. Dans cette thèse, on étudie les groupes de Cremona en considérant certaines classes de „grands'' sous-groupes. Dans la première partie on considère des plongements algébriques de Cr_2(C) vers Cr_n(C). On décrit notamment quelques propriétés géométriques d'un plongement de Cr_2(C) dans Cr_5(C) dû à Gizatullin. En outre, on classifie tous les plongements algébriques de Cr_2(C) dans Cr_3(C) et on généralise ce résultat partiellement pour les plongements de Cr_n(C) dans Cr_{n+1}(C). Dans la deuxième partie, on regarde les suites des degrés des transformations birationnelles des variétés définies sur un corps quelconque. On montre qu'il n'existe qu'un nombre dénombrable de telles suites et on donne de nouvelles contraintes sur la croissance des degrés des automorphismes de l'espace affine de dimension n. Dans la troisième partie, on classifie les sous-groupes de Cr_2(C) qui ne contiennent que des éléments elliptiques, c'est-`a-dire des éléments dont les degrés des itérés sont bornés. On en déduit notamment l'alternative de Tits pour les sous-groupes quelconques de Cr_2(C). Dans la dernière partie on montre que tous les sous-groupes simples de type fini de Cr_2(C) sont finis et, sous l'hypothèse d'un lemme conjectural, qu'un groupe simple se plonge dans Cr_2(C) si et seulement s'il se plonge dans PGL_3(C). / The Cremona group in n-variables Cr_n(C) is the group of birational transformations of the complex projective n-space. This thesis contributes to the research on Cremona groups through the study of certain classes of „large'' subgroups. In the first part we consider algebraic embeddings of Cr_2(C) into Cr_n(C). In particular, we describe geometrical properties of an embedding of Cr_2(C) into Cr_5(C) that was discovered by Gizatullin. We also classify all algebraic embeddings from Cr_2(C) into Cr_3(C), and we partially generalize this result to embeddings of Cr_n(C) into Cr_{n+1}(C). In a second part, we look at degree sequences of birational transformations of varieties over arbitrary fields. We show that there exist only countably many such sequences and we give new obstructions on the degree growth of automorphisms of affine n-space. In the third part, we classify subgroups of Cr_2(C) containing only elliptic elements, i.e. elements whose iterates are of bounded degree. From this we deduce in particular the Tits alternative for arbitrary subgroups of Cr_2(C). In the last part, we show that every finitely generated simple subgroup of Cr_2(C) is finite and, under the hypothesis of an unproven conjectural lemma, that a simple group can be embedded into Cr_2(C) if and only if it can be embedded into PGL_3(C).

Dans cette thèse nous commençons par décrire le groupe des automorphismes extérieurs du groupe des automorphismes polynomiaux du plan affine : il s'identifie au groupe des automorphismes du corps des complexes. Nous étendons ce résultat au groupe de Cremona ; les techniques utilisées sont différentes, le premier groupe ayant une structure de produit amalgamé ce qui n'est pas connu pour le second. Ensuite nous nous intéressons aux représentations de certains réseaux dans le groupe de Cremona ; nous obtenons un théorème de rigidité pour SL(3,Z) et des obstructions à ce que certains réseaux se plongent dans le groupe de Cremona. Enfin nous exhibons une famille de transformations birationnelles curieuses : bien qu'elles présentent toutes les caractéristiques des transformations réputées sans dynamique les expériences numériques révèlent des orbites chaotiques situées dans le complément de deux zones où les adhérences des orbites sont des tores ou des cercles.

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Le attività di ricerca e catalogazione effettuate durante lo stage condotto tra il 2010 e il 2011 sui materiali fittili conservati nel deposito del Museo d’Arte Antica del Castello Sforzesco di Milano sono state all’origine di alcune riflessioni e considerazioni sul tema delle terrecotte decorative del primo Rinascimento in Lombardia - in particolar modo a Milano e a Cremona - che ha trovato un primo importante esito nella presentazione al convegno del 2011 Terrecotte nel Ducato di Milano. Artisti e cantieri del primo Rinascimento. La scelta, in seno alle ricerche per il dottorato, è stata quella di proseguire, incrementandoli, gli studi sulla collezione fittile delle Civiche Raccolte d’Arte di Milano, approfondendo il tema della produzione coroplastica decorativa nell’età sforzesca, non tralasciando di analizzare, per alcuni casi rilevanti, il coevo contesto architettonico. I risultati ottenuti, riordinati e raccolti nelle pagine di questa tesi, costituiscono il tentativo di una prima mappatura dei repertori ornamentali, dove partendo dai patterns decorativi esibiti dalle singole formelle presenti nel deposito del museo è stato possibile stabilire confronti e tracciare relazioni con i numerosi complessi architettonici con decorazione fittile sparsi sul territorio lombardo e nelle regioni confinanti. / The research and cataloguing activities conducted during the internship in 2010-2011 on the clay materials preserved in the deposit of the Museo d’Arte Antica at the Castello Sforzesco in Milan, were what sparked off a series of considerations and reflections on the subject of decorative terracottas during the early Renaissance in Lombardy - particularly Milan and Cremona - the results of which were discussed during the presentation at a conference held in 2011 entitled Terrecotte nel Ducato di Milano. Artisti e cantieri del primo Rinascimento. The decision behind the research conducted for my PhD was to continue and extend the studies to include the collection of clay products preserved by the Art Collections of the Municipality of Milan, more specifically the decorative coroplast productions in the Sforzesca era, which included an in-depth analysis of the contemporary architectonic context of some of the most significant cases. The results obtained, indexed and included in this thesis aim to present an initial mapping of the ornamental repertoires where, starting from the decorative patterns of the individual tiles preserved in the deposit of the Museum, it was possible to establish comparisons and trace relations with the countless architectonic complexes highlighting clay decorations scattered throughout the Lombardy region and bordering areas.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation, we will present a projective parameterization class that resemble the classic maps of Jonquières. We will have the aim to show how the main properties of your ideal base, such as the structure of sizígias and the free presentation of this ideal. In this sense, we obtain the implicit equation that defines the your image and we explicit the formula the degree of the equation. Finally, we will determine Rees algebra equations associated with the base ideal of the parameterization, using computational methods. This parameterization, called parameterization of Jonquières, is constructed from a Cremona and defines a birational map P^n in a hypersurface W of P^n+1. / Nesta dissertação apresentaremos uma classe de parametrizações projetivas que se assemelham aos mapas clássicos de Jonquières e teremos como objetivos mostrar as principais propriedades do seu ideal base, tais como a estrutura das sizígias e a apresentação livre desse ideal, consequentemente obteremos a equação implícita que define a sua imagem, bem como explicitar a fórmula do grau dessa equação. Por fim, determinaremos as equações da álgebra de Rees associada ao ideal base da parametrização, com o uso de métodos computacionais. Tal parametrização, denominada parametrização de Jonquières, é construída a partir de uma Cremona e define um mapa birracional de P^n em uma hipersuperfície W de P^{n+1}.

We establish the existence of a symmetry within the Gromov-Witten theory of CPn and its blowup along points. The nature of this symmetry is encoded in the Cremona transform and its resolution, which lives on the toric variety of the permutohedron. This symmetry expresses some difficult to compute invariants in terms of others less difficult to compute. We focus on enumerative implications; in particular this technique yields a one line proof of the uniqueness of the rational normal curve. Our method involves a study of the toric geometry of the permutohedron, and degeneration of Gromov-Witten invariants. / En este trabajo establecemos la existencia de una simetra en el marco de la teora de Gromov-Witten para CPn y su explosion a lo largo de puntos. La naturaleza de esta simetra queda codicada en la transformacion de Cremona y su resolucion en una variedad torica del permutoedro. Esta simetra expresa algunos invariantes difciles de calcular junto con otros que no lo son tanto. Nos centramos en implicaciones enumerativas; en particular esta tecnica ofrece una prueba enuna lnea de la unicidad de la curva racional normal. Nuestro metodo involucra un estudio de la geometra torica del permutoedro, as como el de la degeneracion de los invariantes de Gromov-Witten.

The project aims to develop a corpus of the 14th century marble sculptures in Brescia, Mantua and Cremona. After mapping the materials localized in these territories, which was made possible thanks to the inventories of the respective Superintendencies and Dioceses, as well as the bibliography, it was possible to collect a large number of testimonies. The resulting archival investigation allowed to refine the research on the history of the pieces, to update and correct the data already reported in bibliography and to investigate unpublished works. From a methodological point of view it was considered appropriate to proceed with the stylistic analysis of the sculptures in parallel with the identification of contexts, origins, authors, models and influences. This made it possible to identify and deepen a critical chapter, the history of Gothic sculpture in this part of Lombardy, which until a few decades ago was rather neglected by studies. The development of a catalog, divided into territorial sections, aims to be an easy reference tool for scholars of Italian sculpture. Several unpublished works have emerged in Brescia and in its province. Some hypotheses have been advanced about the path of the sculptor Delaido da Lodi, active in 1301 in Gargnano, and about the artworks in and from Brescia realized by the Master of Sant’Anastasia. The Mantua area, which returned the most significant results, was analyzed from the beginning of the fourteenth century to the eve of the dalle Masegne season. During this period of time, various artists, mainly from Venice, were active in the city. Among them, the sculptor Andrea da San Felice, the anonymous artist baptized here with the name of "Maestro di Piero Maser" and Antonio da Mestre. Some of the sculptures collected can be related to Lombard artists such as the "Maestro di Viboldone", to whom a new sculpture is attributed, and Guido Frisoni, an artist mentioned in some documents preserved in the State Archive of Mantua, to whom it is now possible to trace the "Madonna with Child" in Grazie di Curtatone, thanks to the interpretation of the epigraph placed at the base. The Cremona area closes the catalogue, which stands out for the quality of the surviving works, at the expense of quantity. The famous reliefs preserved in the church of San Bassiano in Pizzighettone are exemplary of the formal refinement of the survivals of this territory. / Il progetto si è posto l'obiettivo di elaborare un corpus delle sculture del XIV secolo in marmo conservate nelle province di Brescia, Mantova e Cremona. Grazie all’iniziale mappatura dei materiali nei singoli territori, realizzata passando in rassegna gli inventari delle rispettive Soprintendenze e Diocesi, oltre alla bibliografia, è stato possibile raccogliere un cospicuo numero di testimonianze. La conseguente indagine archivistica ha consentito di perfezionare le ricerche sulla storia conservativa dei singoli pezzi, di aggiornare e correggere i dati già segnalati in bibliografia e di indagare sugli inediti. A ciò si è aggiunta l’analisi stilistica delle sculture, che è avvenuta parallelamente all’individuazione di contesti, provenienze, autori, modelli ed influenze. Ciò ha permesso di individuare e fare il punto su un capitolo critico, la storia della scultura gotica in questi territori della Lombardia, che fino a pochi decenni fa era poco frequentato dagli studi. Numerosi sono, inoltre, gli spunti di ricerca emersi per il futuro. Per questo motivo è stato elaborato un catalogo, che ha l’obiettivo di essere uno strumento di agile consultazione per gli studiosi di scultura italiana. Sono emerse diverse opere inedite nel bresciano, su cui si è cercato di far luce. Alcune ipotesi sono state avanzate circa il percorso dello scultore Delaido da Lodi, attivo nel 1301 a Gargnano, e riguardo alle testimonianze bresciane del Maestro di Sant’Anastasia. Il territorio mantovano, che ha restituito i risultati più significativi, è stato analizzato dagli esordi del Trecento fino alla vigilia della stagione dei dalle Masegne. Durante questo lasso di tempo furono attivi in città diversi artisti, principalmente provenienti da Venezia, su cui ora è possibile ragionare. Tra essi emergono lo scultore Andrea da San Felice, l'anonimo artista battezzato in questa sede con il nome di "Maestro di Piero Maser" e Antonio da Mestre. Di origine lombarda furono invece il Maestro delle sculture di Viboldone, a cui in questa sede è attribuita una nuova opera, e Guido Frisoni da Como, artista citato in alcuni documenti conservati a Mantova, in Archivio di Stato, a cui è ora possibile ricondurre la 'Madonna con Bambino' di Grazie di Curtatone grazie all'interpretazione dell'epigrafe posta alla base dell'opera. Chiude il lavoro il territorio di Cremona, che si distingue per la qualità delle opere superstiti, a discapito della quantità. Esemplificativi della ricercatezza formale delle sopravvivenze di questo territorio sono i celebri rilievi che si conservano nella chiesa di San Bassiano a Pizzighettone.

(in English) The key theme of my thesis are venetian influences on the Boccaccio Boccaccino's work, who was one of the most important exponents of the Cremonese school of painters. Initially he worked in Genoa, Cremona and Milan and he was influenced by the painters as Leonardo, Bramantino and Boltraffio. In the years 1497-1500 Boccaccino is documented in Ferrara. In this period he executed so-called tondo Gronau, The Christ on the way to Calvary, The Virgin and Child, now in Boston, The Virgin and Child, now in Padua, The Adoration of the Shepherds, now in Naples and Dead Christ supported by an Angel. This works show the influence of Bramantino, umbrian school but also early influence of venetian art. In 1500 or 1501 he painted the altarpiece with Virgin and Child with Sts Peter, Michael, John the Baptist and John the Evangelist for the church of S. Giuliano in Venice. Models of this composition are the S. Cassiano altarpiece of Antonello da Messina and Virgin and Child with Saints which Giovanni Bellini executed for the church of S. Giobbe. Boccaccino's image in S.Giuliano is also inspired by Ercole de'Roberti and Lorenzo Costa. His colours show the influence of Giorgione. In 1506 is Boccaccino documented in Venice but also in Cremona. In the years 1500-1506 he stayed probably in Venice, but he...

Ce travail peut être divisé en trois partie: 1. Théorie des groupes. Il s'agit ici d'une étude de la structure du groupe T de Thompson. On explique la notion de la mutation linéaire par morceaux et on obtient la nouvelle présentation de ce groupe en termes des génerateurs et relations. 2. Géometrie birationnelle. On étudie en détail le groupe de Cremona qui est un groupe des automorphismes birationnels du plan projectif. En particulier on s'interesse à son sous-groupe Symp des elements qui préserve le crochet de Poisson dit logarithmique, aussi bien qu'à un sous-groupe H engendré par SL(2,Z) et par les mutations. On construit des limites projectives des surfaces sur lesquelles ces groupes agissent régulièrement, et on en déduit les répresentations linéaires de ces groupes dans les limites inductives des groupes de Picard des surfaces. 3. Algèbre homologique. A partir d'une variété algébrique on construit une catégorie triangulée qui ne dépend que de sa classe birationnelle. En utilisant la technique de quotient de dg-catégories, on calcule explicitement cette catégorie pour les surfaces rationnelles. Comme consequence on obtient l'action du groupe de Cremona sur une algébre non-commutative par les automorphismes extérieures. On donne les applications de ces résultats aux formules des mutations des variables non-commutatives.