Thèse--Lettres--Montpellier, 1961. / Bibliogr. p. 277-288.

Th.--Lettres--Paris, 1910.

Le foncteur de Picard d'un schéma a fait l'objet d'une étude approfondie dans les années soixante. La décennie suivante a vu naître avec les travaux de Giraud puis Deligne, Mumford, et enfin Artin la notion de champ algébrique, qui généralise celle de schéma. Nous nous intéressons dans cette thèse au foncteur de Picard d'un champ algébrique et démontrons à son sujet un certain nombre de résultats bien connus dans le cadre des schémas. Nous étudions entre autres la représentabilité du foncteur de Picard, ses propriétés de séparation, de finitude relative, et les déformations de faisceaux inversibles. Nous construisons également la composante neutre du foncteur de Picard et étudions sa propreté. Quelques exemples viennent étayer le propos. Ces travaux nous ont amené à résoudre un certain nombre de problèmes techniques relatifs à la cohomologie des faisceaux abéliens sur le site lisse-étale d'un champ algébrique. Ces questions ont été rassemblées en annexe en fin de volume.

The main goal of the thesis is to give explicit formulas for the computation of the local Picard group of some binoid algebras. In particular the Stanley-Reisner and the general monomial case are covered. In order to do so, we introduce a new topology on the spectrum of the binoid algebra over a field, that we call combinatorial topology, coarser than Zariski topology, that mimics the topology on the spectrum of a pointed monoid. Then we use the tools of cohomology of sheaves on schemes of pointed monoids in order to prove some formulas for computing the cohomology of the sheaf of units on the punctured spectrum of a simplicial pointed monoid. We prove that the Picard group of a Stanley-Reisner algebra is trivial with the Zariski topology and, finally, we use these results and the combinatorial topology to obtain the explicit formulas. Lastly, we extend this result to any quotient of the polynomial ring over a monomial ideal.

L’héritage littéraire de Picard s’élève à une centaine des pièces de théâtre et environ six romans. Il y traite les problèmes sociaux, politiques et religieux de son époque, à l’instar de Molière qui semble être la première source d’inspiration de notre dramaturge. Dans ses Mémoires, Alexandre Dumas décrit Louis-Benoît Picard comme « le petit Molière du XIXe siècle ou le moderne Molière ». Ce rapprochement paraît plus ou moins discutable dans la mesure où, de nos jours, les traces de ce « petit Molière » sont rares et il est désormais totalement oublié par l’histoire littéraire. Entre cette affirmation de Dumas et la place médiocre qu’occupe notre dramaturge dans l’histoire du théâtre, deux explications pourraient être envisagées. La première réside dans le fait que la période révolutionnaire où Louis-Benoît Picard a vécu a été trop chargée historiquement et les critiques qui s’intéressent à cette période accordent plus d’importance aux événements historiques qu’aux productions littéraires, de sorte que la plupart des écrivains de cette période sont tombés dans l’oubli. La deuxième explication revient aux productions littéraires de Picard elles-mêmes, considérées comme alourdies de répétitions, conventionnelles et dépourvues de toute originalité. / The literary heritage of Picard can reach as many as a hundred plays and about six novels. He deals with the social, political and religious problems of his time, like Molière, who seems to be the first source of inspiration for our playwright. In his Mémoires, Alexandre Dumas describes Louis-Benoît Picard as "the little Moliere of the nineteenth century or the modern Moliere". This rapprochement seems more or less debatable insofar as the traces of this "little Moliere" are rare today and it is now completely forgotten in Literary History. Between this assertion of Dumas and the mediocre place occupied by our dramatist in the history of the theater, two explanations could be envisaged. The first is that the revolutionary period in which Louis-Benoît Picard lived has been overloaded historically and critics who are interested in this period place more importance on historical events than on literary productions, Most of the writers of this period have fallen into oblivion. The second explanation belongs to Picard's literary productions themselves, represented as repetitive, conventional, and far from original.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 114-117. / by Hans Kleppe. / Ph.D.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2013. / Continued fractions have been extensively studied in number theoretic ways. These continued fractions are expressed as compositions of M¨obius maps in the Picard group PS L(2;C) that act, by Poincar´e’s extension, as isometries on H3. We investigate the Picard group with its generators and derive the fundamental domain using a direct method. From the fundamental domain, we produce an ideal octahedron, O0, that generates the Farey tessellation of H3. We explore the properties of Farey neighbours, Farey geodesics and Farey triangles that arise from the Farey tessellation and relate these to Ford spheres. We consider the Farey addition of two rationals in R as a subdivision of an interval and hence are able to generalise this notion to a subdivision of a Farey triangle with Gaussian Farey neighbour vertices. This Farey set allows us to revisit the Farey triangle subdivision given by Schmidt [44] and interpret it as a theorem about adjacent octahedra in the Farey tessellation of H3. We consider continued fraction algorithms with Gaussian integer coe cients. We introduce an analogue of Series [45] cutting sequence across H2 in H3. We derive a continued fraction expansion based on this cutting sequence generated by a geodesic in H3 that ends at the point in C that passes through O0.

We study the Brauer classes rising from the obstruction to the existence of tautological line bundles on the Picard scheme of curves. We establish various properties of the Brauer classes for families of smooth curves. We compute the period and index of the Brauer class associated with the universal smooth curve for a fixed genus. We also show such Brauer classes are trivialized when we specialize to certain generalized theta divisors. If we consider the universal totally degenerate curve with a fixed dual graph, using symmetries of the graph, we give bounds on the period and index of the Brauer classes. As a result, we provide some division algebras of prime degree, serving as candidates for the cyclicity problem. As a byproduct, we re-calculate the period and index of the Brauer class for universal smooth genus g curve in an elementary way. We study certain conic associated with the universal totally degenerate curve with a fixed dual graph. We show the associated conic is non-split in some cases. We also study some other related geometric properties of Brauer groups.

Thesis (doctoral)--Universität Zürich, 1905.

In a series of works, Jun-Muk Hwang and Ngaiming Mok have developed a geometric theory of uniruled projective manifolds, especially those of Picard Number 1, relying on the study of Varieties of Minimal Rational Tangents (VMRT) from both the algebro-geometric and the G-structure perspectives. Based on this theory, Ngaiming Mok and Jaehyun Hong studied the standard embedding between two Rational Homogeneous Spaces (RHS) associated to long simple roots which are of different dimensions. In this thesis, I consider admissible pairs of RHS (X0, X) of Picard number 1 and locally closed complex submanifolds S ⊂ X inheriting VMRT sub-structures modeled on X0 = G0/P0 ⊂ X = G/P de_ned by taking intersections of VMRT of X with tangent space of S. Moreover, if any such S modeled on (X0, X) is necessarily the image of a standard embedding i : X0 → X, (X0, X) is said to be rigid. In this thesis, it is proved that an admissible pair (X0, X) is rigid whenever X is associated to a long simple root and X0 is non-linear and de_ned by a marked Dynkin sub-diagram. In the case of the pair (S0, S) of compact Hermitian Symmetric Spaces (cHSS), all the admissible pairs (S0, S) are completely classified. Based on this classification, a sufficient condition for the pair (S0, S) to be non-rigid is established through explicitly constructing a submanifold S ⊂ S such that S can never be obtained from the image of any standard embedding i : S0 → S. Besides, the term special pair is coined for those (S0; S) sorted out through classification, and the algebraicity of submanifolds modeled on special pairs is confirmed by checking a modified form of the non-degeneracy condition defined by Hong and Mok is satisfied. However, the question as to whether these special pairs are rigid, as pointed out in this thesis, remains to be investigated. Finally, pairs of hyperquadrics (Q^n, Q^m) are studied separately. Since non-rigidity is trivial, in these cases it is interesting to establish a characterization of the standard embedding i : Q^n→Q^m under some stronger condition. In this thesis, the latter problem is solved in terms of the partial vanishing of second fundamental forms. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy