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In this dissertation, we study morphisms of smooth complex projective varieties to the projective line with at most two singular fibres. We show that if such a morphism has at most one singular fibre, then the domain of the morphism is uniruled and the morphism admits algebraic sections. We reach the same conclusions, but with algebraic genus zero multisections instead of algebraic sections, if the morphism has at most two singular fibres and the first Chern class of the domain of the morphism is supported in a single fibre of the morphism. To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon's virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections.

The family of algebraic surfaces X dened by the single equation zn = (y a1x) (y anx)(x 1) over an algebraically closed eld k of characteristic zero, where a1; : : : ; an 2 k are distinct, is studied. It is shown that this is a rational surface with a non-rational singularity at the origin. The ideal class group of the surface is computed. The terms of the Chase-Harrison-Rosenberg seven term exact sequence on the open complement of the ramication locus of X ! A2 are computed; the Brauer group is also studied in this unramied setting. The analysis is extended to the surface eX obtained by blowing up X at the origin. The interplay between properties of eX , determined in part by the exceptional curve E lying over the origin, and the properties of X is explored. In particular, the implications that these properties have on the Picard group of the surface X are studied. / by Drake Harmon. / Vita. / Thesis (Ph.D.)--Florida Atlantic University, 2013. / Includes bibliography. / Mode of access: World Wide Web. / System requirements: Adobe Reader.

Chapter 0 In this introductory chapter, certain notational and terminological conventions are established and a summary given of background results that are needed in subsequent chapters. Chapter 1 In this chapter, the notion of a "weak conguence formula" [Tay72], [BB75] is introduced and used to characterize both subdirectly irreducible algebras and essential extensions. Special attention is paid to the role they play in varieties with definable principal congruences. The chapter focuses on residually small varieties; several of its results take their motivation from the so-called "Quackenbush Problem" and the "RS Conjecture". One of the main results presented gives nine equivalent characterizations of a residually small variety; it is largely due to W. Taylor. It is followed by several illustrative examples of residually small varieties. The connections between residual smallness and several other (mostly categorical) properties are also considered, e.g., absolute retracts, injectivity, congruence extensibility, transferability of injections and the existence of injective hulls. A result of Taylor that establishes a bound on the size of an injective hull is included. Chapter 2 Beginning with a proof of A. Day's Mal'cev-style characterization of congruence modular varieties [Day69] (incorporating H.-P. Gumm's "Shifting Lemma"), this chapter is a self-contained development of commutator theory in such varieties. We adopt the purely algebraic approach of R. Freese and R. McKenzie [FM87] but show that, in modular varieties, their notion of the commutator [α,β] of two congruences α and β of an algebra coincides with that introduced earlier by J. Hagemann and C. Herrmann [HH79] as well as with the geometric approach proposed by Gumm [Gum80a],[Gum83]. Basic properties of the commutator are established, such as that it behaves very well with respect to homomorphisms and sufficiently well in products and subalgebras. Various characterizations of the condition "(x, y) Є [α,β]” are proved. These results will be applied in the following chapters. We show how the theory manifests itself in groups (where it gives the familiar group theoretic commutator), rings, modules and congruence distributive varieties. Chapter 3 We define Abelian congruences, and Abelian and affine algebras. Abelian algebras are algebras A in which [A2, A2] = idA (where A2 and idA are the greatest and least congruences of A). We show that an affine algebra is polynomially equivalent to a module over a ring (and is Abelian). We give a proof that an Abelian algebra in a modular variety is affine; this is Herrmann's Funda- mental Theorem of Abelian Algebras [Her79]. Herrmann and Gumm [Gum78], [Gum80a] established that any modular variety has a so-called ternary "difference term" (a key ingredient of the Fundamental Theorem's proof). We derive some properties of such a term, the most significant being that its existence characterizes modular varieties. Chapter 4 An important result in this chapter (which is due to several authors) is the description of subdirectly irreducible algebras in a congruence modular variety. In the case of congruence distributive varieties, this theorem specializes to Jόnsson's Theorem. We consider some properties of a commutator identity (Cl) which is a necessary condition for a modular variety to be residually small. In the main result of the chapter we see that for a finite algebra A in a modular variety, the variety V(A) is residually small if and only if the subalgebras of A satisfy (Cl). This theorem of Freese and McKenzie also proves that a finitely generated congruence modular residually small variety has a finite residual bound, and it describes such a bound. Thus, within modular varieties, it proves the RS Conjecture. Conclusion The conclusion is a brief survey of further important results about residually small varieties, and includes mention of the recently disproved (general) RS Conjecture. / Thesis (M.Sc.)-University of Natal, Durban, 2000.

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 132-136).

Serge Ovsienko provou que a variedade de Gelfand-Tsetlin para gl(n) é equidimensional (i.e., todas suas componentes irredutíveis têm a mesma dimensão) com dimensão n(n-1)/2. Este resultado é conhecido como \"Teorema de Ovsienko\" e tem importantes consequências na Teoria de Representacões de Álgebras. Neste trabalho, provamos uma versão fraca do Teorema de Ovsienko para gl(n) e estendemos tal versão fraca a uma estrutura que tem como caso particular gl(3), esse é o caso do grupo quântico Yangian Yp(gl(3)) de nível p. Além disso, o Teorema de Ovsienko também tem consequências na Geometria Simplética, especificamente na equidimensionalidade das fibras em uma projeção da aplicação de Kostant-Wallach. Neste trabalho apresentamos a generalização deste resultado. / Serge Ovsienko proved that the Gelfand-Tsetlin variety for gl(n) is equidimensional (i.e., all its irreducible components have the same dimension) with dimension n(n-1)/2. This result is known as \"Ovsienko\'s Theorem\" and it has important consequences in Representation Theory of Algebras. In this work, we prove a weak version of Ovsienko\'s Theorem for gl(n) and we extend that weak version to a structure which has as particular case gl(3), this case is the quantum group level p Yangian Yp(gl(3)). Moreover, the theorem of Ovsienko also has consequences in Symplectic Geometry, more concretely in the equidimensionality of the fibers in a projection of the Kostant-Wallach map. In this work we will present the generalization of that result.

We consider systems of polynomial equations and inequalities to be solved in integers. By applying the circle method, when the number of variables is large and the system is geometrically well-behaved we give an asymptotic estimate for the number of solutions of bounded size. In the case of R homogeneous equations having the same degree d, a classic theorem of Birch provides such an estimate provided the number of variables is R(R+1)(d-1)2^d-1+R or greater and the system is nonsingular. In many cases this conclusion has been improved, but except in the case of diagonal equations the number of variables needed has always grown quadratically in R. We give a result requiring only d2^dR+R variables, obtaining linear growth in R. When d = 2 or 3 we require only that the system be nonsingular; when d<4 we require that the coefficients of the equations belong to a certain explicit Zariski open set. These conditions are satisfied for typical systems of equations, and can in principle be checked algorithmically for any particular system. We also give an asymptotic estimate for the number of solutions to R polynomial inequalities of degree d with real coefficients, in the same number of variables and satisfying the same geometric conditions as in our work on equations. Previously one needed the number of variables to grow super-exponentially in the degree d in order to show that a nontrivial solution exists.

Serge Ovsienko provou que a variedade de Gelfand-Tsetlin para gl(n) é equidimensional (i.e., todas suas componentes irredutíveis têm a mesma dimensão) com dimensão n(n-1)/2. Este resultado é conhecido como \"Teorema de Ovsienko\" e tem importantes consequências na Teoria de Representacões de Álgebras. Neste trabalho, provamos uma versão fraca do Teorema de Ovsienko para gl(n) e estendemos tal versão fraca a uma estrutura que tem como caso particular gl(3), esse é o caso do grupo quântico Yangian Yp(gl(3)) de nível p. Além disso, o Teorema de Ovsienko também tem consequências na Geometria Simplética, especificamente na equidimensionalidade das fibras em uma projeção da aplicação de Kostant-Wallach. Neste trabalho apresentamos a generalização deste resultado. / Serge Ovsienko proved that the Gelfand-Tsetlin variety for gl(n) is equidimensional (i.e., all its irreducible components have the same dimension) with dimension n(n-1)/2. This result is known as \"Ovsienko\'s Theorem\" and it has important consequences in Representation Theory of Algebras. In this work, we prove a weak version of Ovsienko\'s Theorem for gl(n) and we extend that weak version to a structure which has as particular case gl(3), this case is the quantum group level p Yangian Yp(gl(3)). Moreover, the theorem of Ovsienko also has consequences in Symplectic Geometry, more concretely in the equidimensionality of the fibers in a projection of the Kostant-Wallach map. In this work we will present the generalization of that result.

Cette thèse est consacrée aux propriétés géométriques des opérations de tores algébriques dans les variétés affines. Elle est issue de trois prépublications qui correspondent aux points (1), (2), (3) ci-après. Soit X une variété affine munie d’une opération d’un tore algébrique T. Nous appelons complexité la codimension de l’orbite générale de T dans X. Sous l’hypothèse de normalité et lorsque le corps de base est algébriquement clos de caractéristique 0, la variété X admet une description combinatoire en termes de géométrie convexe. Cette description, obtenue en 2006 par Altmann et Hausen, généralise celle classique des variétes toriques. Notre but consiste à étudier des problèmes nouveaux concernant les propriétés algébriques et géométriques de X lorsque l’operation de T dans X est de complexité 1. (1) Dans la première partie, un résultat donne une manière explicite de déterminer la clôture intégrale de toute variété affine définie sur un corps algébriquement clos de caractérisque 0 munie d’une opération de T de complexité 1 en termes de la description combinatoire d’Altmann-Hausen. Comme application, nous donnons une classification complète des idéaux intégralement clos homogènes de l’algèbre des fonctions régulières de X et généralisons un théorème de Reid-Roberts-Vitulli sur la description de certains idéaux normaux de l’algèbre des polynômes à plusieurs variables. (2) Les calculs de la première partie suggèrent une démonstration de la validité de la présentation d’Altmann-Hausen sur un corps quelconque dans le cas de complexité 1. Ce qui est fait dans la deuxième partie. Dans la situation non déployée, la descente galoisienne d’une variété affine normale munie d’une opération d’un tore algébrique de complexité 1 est décrite par un nouvel objet combinatoire que nous appelons diviseur polyédral Galois stable. (3) Dans la troisième partie, lorsque que le corps de base est parfait, nous classifions toutes les opérations du groupe additif dans X normalisées par l’action de T de complexité 1. Cette classification généralise des travaux classiques de Flenner et Zaidenberg dans le cas des surfaces et de Liendo dans le cas où le corps ambiant est algébriquement clos de caractéristique 0. / This thesis is devoted to the study of geometric properties of affine algebraic varieties endowed with an action of an algebraic torus. It comes from three preprints which correspond to the indicated points (1), (2), (3). Let X be an affine variety equipped with an action of the algebraic torus T. The complexity of the T-action on X is the codimension of the general T-orbit. Under the assumption of normality and when the ground field is algebraically closed of characteristic 0, the variety X admits a combinatorial description in terms of convex geometry. This description obtained by Altmann and Hausen in the year 2006 generalizes the classical one for toric varieties. Our purpose is to investigate new problems on the algebraic and geometric properties of the variety X when the T-action on X is of complexity 1. (1) In the first part, a result gives an effective method to determine the integral closure of any affine variety defined over an algebraically field of characteristic 0 with a T-action of complexity 1 in terms of the combinatorial description of Altmann-Hausen. As an application, we provide an entire classification of the homogeneous integrally closed ideals of the algebra of regular functions on X and generalize the Reid-Roberts-Vitulli's theorem on the description of certain normal ideals of the polynomial algebra. (2) The calculations of the first part suggest a proof of the validity of the presentation of Altmann-Hausen in the case of complexity 1 over an arbitrary ground field. This is done in the second part of this thesis. In the non-split situation, the Galois descent of normal affine varieties with a T-action of complexity 1 is described by a new combinatorial object which we call a Galois invariant polyhedral divisor. (3) In the third part, when the base field is perfect, we classify all the actions of the additive group on X normalized by the T-action of complexity 1. This classification generalizes classical works of Flenner and Zaidenberg in the surface case and of Liendo when the base field is algebraically closed of characteristic 0.

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