Th. 3e cycle--Math. pures--Grenoble 1, 1982. N°: 7.

Traditional algebraic geometric invariants lose some of their potency in positive characteristic. For instance, smooth projective hypersurfaces may be covered by lines despite being of arbitrarily high degree. The purpose of this dissertation is to define a class of hypersurfaces that exhibits such classically unexpected properties, and to offer a perspective with which to conceptualize such phenomena. Specifically, this dissertation proposes an analogy between the eponymous q-bic hypersurfaces—special hypersurfaces of degree q+1, with q any power of the ground field characteristic, a familiar example given by the corresponding Fermat hypersurface—and low degree hypersurfaces, especially quadrics and cubics. This analogy is substantiated by concrete results such as: q-bic hypersurfaces are moduli spaces of isotropic vectors for a bilinear form; the Fano schemes of linear spaces contained in a smooth q-bic hypersurface are smooth, irreducible, and carry structures similar to orthogonal Grassmannian; and the intermediate Jacobian of a q-bic threefold is purely inseparably isogenous to the Albanese variety of its smooth Fano surface of lines.

by Wong Sai Yiu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 94-97). / Introduction --- p.iii / Chapter 1 --- Invariants on ideals of holomorphic function germs --- p.1 / Chapter 1.1 --- Preliminaries --- p.1 / Chapter 1.2 --- Ideals of holomorphic function germs --- p.3 / Chapter 1.3 --- The order of contact of an ideal --- p.7 / Chapter 1.4 --- Higher order invariants --- p.11 / Chapter 2 --- Geometry on real hypersurfaces of Cn --- p.14 / Chapter 2.1 --- CR geometry --- p.14 / Chapter 2.2 --- The associated family of holomorphic ideals on real subvaxiety of Cn --- p.18 / Chapter 2.3 --- Relationships between points of finite type and complex varieties --- p.25 / Chapter 2.4 --- The case of pseudoconvex real hypersurfaces --- p.33 / Chapter 2.5 --- Other finite type conditions --- p.35 / Chapter 3 --- Point of finite type and the d-Neumann problem --- p.44 / Chapter 3.1 --- Introduction --- p.44 / Chapter 3.2 --- Subellipticity and subelliptic multipliers --- p.47 / Chapter 3.3 --- Geometry on Kohn's ideals of subelliptic multipliers --- p.60 / Chapter 3.4 --- The Diederich - Fornaess theorem --- p.66 / Chapter 3.5 --- Catlin's necessary condition on subellipticity --- p.69 / Chapter 4 --- Analysis on finite type domains --- p.78 / Chapter 4.1 --- The Bergman projection --- p.78 / Chapter 4.2 --- Boundary regularity of proper holomorphic mappings --- p.83 / Chapter 4.3 --- Local regularity and extension of CR mappings --- p.88 / Bibliography --- p.94

Lo, Chi Yu. / On t.p. "n" is a superscript. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 92). / Abstracts in Chinese and English. / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- "CR structures and the group SU(p, q)" --- p.7 / Chapter 1.1 --- Almost complex structure and CR manifolds --- p.7 / Chapter 1.2 --- Automorphism Groups of Ball and Polydisc --- p.17 / Chapter 1.3 --- "The group SU(p,q) and its Maurer Cartan form" --- p.24 / Chapter 2 --- Cartan´ةs construction on nondegenerate CR manifold --- p.33 / Chapter 2.1 --- A digression on the Frobenius Theorem and projective structure --- p.33 / Chapter 2.2 --- Cartan bundle and canonical forms --- p.45 / Chapter 2.3 --- Calculations of real hypersurface in C2 --- p.60 / Chapter 3 --- Geometric consequences and chain --- p.66 / Chapter 3.1 --- CR equivalence problem --- p.66 / Chapter 3.2 --- CR manifolds of dimension 3 --- p.71 / Chapter 3.3 --- Definition of chains --- p.78 / Chapter 3.4 --- Chains on a special kind of Reinhardt hyper surf ace in C2 --- p.87 / Bibliography --- p.92

Hui, Chun Yin. / On t.p. "n" is a superscript. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 80-83). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Basic properties of real hypersurfaces in CN --- p.9 / Chapter 2.1 --- Hypersurfaces in CN and some nondegeneracy conditions --- p.9 / Chapter 2.2 --- CR functions and their holomorphic extensions --- p.15 / Chapter 2.3 --- Normal coordinates for real analytic hypersurfaces --- p.18 / Chapter 3 --- The algebraic results for reflection principle --- p.22 / Chapter 4 --- Reflection principle for real analytic hypersurfaces in higher complex dimensions --- p.30 / Chapter 4.1 --- Reflection principle for Levi nondegenerate hypersurfaces --- p.31 / Chapter 4.2 --- Essentially finite real analytic hypersurfaces and not totally degenerate CR mappings --- p.38 / Chapter 4.3 --- Reflection principle for essentially finite hypersurfaces --- p.44 / Chapter 4.4 --- Reflection principle for CR mappings and bounded domains --- p.54 / Chapter 4.5 --- Futher results on the reflection principle --- p.64 / Chapter 5 --- An extension result of CR functions by a general Schwarz reflection principle --- p.66 / Chapter 5.1 --- A general Schwarz reflection principle --- p.66 / Chapter 5.2 --- "Holomorphic extension of CR functions on a real analytic, generic CR submanifold in CN" --- p.69 / Bibliography --- p.80

Thesis (Ph.D.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2009/06/08). Includes bibliographical references.

Thesis (Ph. D.)--University of Washington, 2006. / Vita. Includes bibliographical references (p. 63-66).

Hypergeometric functions seem to be ubiquitous in mathematics. In this document, we present a couple of ways in which hypergeometric functions appear in arithmetic geometry. First, we show that the number of points over a finite field [mathematical symbol] on a certain family of hypersurfaces, [mathematical symbol] ([lamda]), is a linear combination of hypergeometric functions. We use results by Koblitz and Gross to find explicit relationships, which could be useful for computing Zeta functions in the future. We then study more geometric aspects of the same families. A construction of Dwork's gives a vector bundle of deRham cohomologies equipped with a connection. This connection gives rise to a differential equation which is known to be hypergeometric. We developed an algorithm which computes the parameters of the hypergeometric equations given the family of hypersurfaces. / text

Thesis (Ph. D.)--University of Texas at Austin, 2009. / Title from PDF title page (University of Texas Digital Repository, viewed on Sept. 9, 2009). Vita. Includes bibliographical references.

Depuis les 50 dernières années, de nombreux progrès ont été faits dans la compréhension du comportement asymptotique du nombre de points rationnels de hauteur bornée sur les variétés algébriques. Des conjectures précises ont été avancées par Baryrev, Manin et Peyre quant à la formule asymptotique attendue pour une variété générale.En 1962, à l'aide d'arguments issus de la méthode du cercle de Hardy et Littlewood, B. Birch a donné une estimation précise du nombre de points à coordonnées entières bornées dans une hypersurface définie par une équation homogène. Ceci revient à démontrer la conjecture de Batyrev-Manin-Peyre pour les hypersurfaces de l'espace projectif. Plus récemment, V. Blomer et J. Brüdern ont élaboré des techniques leur permettant d'établir une formule pour le comportement asymptotique du nombre de points de hauteur bornée pour des hypersurfaces d'espaces multiprojectifs définies par des équations multihomogènes diagonales. Parallèlement, D. Schindler a démontré la conjecture pour des hypersurfaces générales d'espaces biprojectifs, à l'aide de développements de la méthode de Birch.L'objet de cette thèse a été d'utiliser et de généraliser les techniques de Schindler, Blomer et Brüdern afin de démontrer la validité de la conjecture de Batyrev-Manin-Peyre pour le cas d'hypersurfaces de variétés toriques plus générales.Ce travail est composé de trois parties. La première partie concerne le cas particulier des hypersurfaces de tridegré (1,1,1) d'un espace triprojectif. Ce cas particulier constitue une première extension des techniques de Schindler à des variétés toriques dont le rang du groupe de Picard est 3. La deuxième partie est consacrée à l'étude des hypersurfaces d'une famille de variétés toriques dont le rang du groupe de Picard est 2 et contenant la famille des espaces biprojectifs. Il s'agit en effet d'étendre la méthode de Schindler afin d'obtenir une formule asymptotique pour le nombre de points de hauteur bornée sur ces variétés. Enfin, dans la dernière partie, nous généralisons les méthodes développées dans les deux parties précédentes à des hypersurfaces des variétés toriques complètes lisses de rang de groupe dont le cône effectif est supposé simplicial, ce qui nous permet de démontrer la conjecture de Batyrev-Manin-Peyre pour ces variétés. / For the last 50 years, many progresses have been made in the understanding of the asymptotic behaviour of the number of rational points of bouded height on algebraic varieties. Some precise conjectures have been advanced by Batyrev, Manin, and Peyre for the expected asymptotic formula for a general variety.In 1962, using some arguments of the Hardy-Littlewood circle method, B. Birch gave a precise estimate for the number of integral points whose coordinates are bounded on an hypersurface defined by an homogeneous equation. This amounts to demonstrating the Batyrev-Manin-Peyre conjecture for hypersurfaces of projective spaces. More recently, V. Blomer and J. Brüdern developed some methods permitting to establish a formula for the asymptotic growth of the number of points of bounded height on hypersurfaces of multiprojective spaces defined by multihomogeneous diagonal equations. In the same time, D. Schindler proved the conjecture for general hypersurfaces of biprojective spaces by using some developements of the method of Birch.The aim of this thesis was to use and generalize the methods of Schindler, blomer, and Brüdern in order to prove the Batyrev-Manin-Peyre conjecture in the case of hypersurfaces of some general toric varieties.This work contain three parts. The first one deals with the particular case of hypersurfaces of tridegree (1,1,1) of triprojective spaces. This particular case is a first extension of the method of Schindler to some toric varieties whose rank of the Picard group is 3. The second part deals with the study of hypersurfaces of a class of toric varieties whose rank of the Picard group is 2 and containing biprojective spaces. We establish a generalization of the method of Schindler method in order to find an asymptotic formula for the number of points of bounded height on these vrieties. Finally, in the last part, we generalize the methods developed in the last two part to treat the case of hypersurfaces of complete non-singular toric vareties whose effective cone is simplicial. This permits to prove the conjecture of batyrev-Manin-Peyre for these varieties.