The global log canonical threshold, algebraic counterpart to Tian's alpha-invariant, plays an important role when studying the geometry of Fano varieties. In particular, Tian showed that Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In recent years Donaldson drafted a programme to precisely determine when a smooth Fano variety X admits a Kahler-Einstein metric. It was conjectured that the existence of such a metric is equivalent to X being K-stable, an algebraic-geometric property. A crucial step in Donaldson's programme consists on finding a Kahler-Einstein metric with edge singularities of small angle along a smooth anticanonical boundary. Jeffres, Mazzeo and Rubinstein showed that a dynamic version of the alpha-invariant could be used to find such metrics. The global log canonical threshold measures how anticanonical pairs fail to be log canonical. In this thesis we compute the global log canonical threshold of del Pezzo surfaces in various settings. First we extend Cheltsov's computation of the global log canonical threshold of complex del Pezzo surfaces to non-singular del Pezzo surfaces over a ground field which is algebraically closed and has arbitrary characteristic. Then we study which anticanonical pairs fail to be log canonical. In particular, we give a very explicit classifiation of very singular anticanonical pairs for del Pezzo surfaces of degree smaller or equal than 3. We conjecture under which circumstances such a classifcation is plausible for an arbitrary Fano variety and derive several consequences. As an application, we compute the dynamic alpha-invariant on smooth del Pezzo surfaces of small degree, where the boundary is any smooth elliptic curve C. Our main result is a computation of the dynamic alpha-invariant on all smooth del Pezzo surfaces with boundary any smooth elliptic curve C. The values of the alpha-invariant depend on the choice of C. We apply our computation to find Kahler-Einstein metrics with edge singularities of angle β along C.

del Pezzo surfaces are isomorphic to either P¹ x P¹ or P² blown up <i>a</i> times, where <i>a</i> ranges from 0 to 8. We will look at lines on del Pezzo surfaces isomorphic to P² blown up <i>a</i> times with <i>a</i> ranging from 0 to 6. We will show that when we count points of bounded height on one of these surfaces, the number of points on lines give us the primary growth order, but the secondary growth order calculates the number of points on the rest of the surface and hence is a better representation of the geometry of the surface.

One of the fundamental problems in Algebraic Geometry is to study solutions to certain systems of polynomial equations in several variables, or in other words, find rational points on a given variety which is defined by equations. In this paper, we discuss the existence of del Pezzo surface of degree 1 and 2 with a unique rational point over any finite field [Special characters omitted.] , and we will give a lower bound on the number of rational points to each q. Furthermore, we will give explicit equations of del Pezzo surfaces with a unique rational point. Also we will discuss the rationality property of the del Pezzo surfaces especially in lower degrees.

del Pezzo surfaces are isomorphic to either P¹ x P¹ or P² blown up <i>a</i> times, where <i>a</i> ranges from 0 to 8. We will look at lines on del Pezzo surfaces isomorphic to P² blown up <i>a</i> times with <i>a</i> ranging from 0 to 6. We will show that when we count points of bounded height on one of these surfaces, the number of points on lines give us the primary growth order, but the secondary growth order calculates the number of points on the rest of the surface and hence is a better representation of the geometry of the surface.

Texte remanié de: Thèse de doctorat--Mathématiques--Université de Paris-Sud, 2005. Titre de soutenance : Principe de Hasse pour les surfaces de Del Pezzo de degré 4. / Mention parallèle de titre ou de responsabilité : Intersections of two quadrics and pencils of curves of genus 1. Introduction en anglais et en français. Bibliogr. p. 209-213.

Cette thèse concerne l'étude des variétés de Fano, qui sont des objets centraux de la classification des variétés algébriques. La première question abordée concerne les variétés de Fano lisses de dimension quatre. On cherche a étudier les potentielles singularités d'un diviseur anticanonique de sorte qu'on puisse les écrire sous une forme locale explicite. En tant qu'étape intermédiaire, on démontre aussi que ces points sont au plus des singularités terminales, c'est-à-dire les singularités les plus proches du cas lisse du point de vue de la géométrie birationnelle. On montre ensuite que ce dernier résultat se généralise en dimension arbitraire en admettant une conjecture de non-annulation de Kawamata.De façon complémentaire, on s¿intéresse à des variétés de Fano de dimension plus petite, mais admettant des singularités. Il s¿agit des surfaces de del Pezzo ayant des singularités de type 1/3(1,1). Ceci est l'exemple le plus simple de singularité rigide, c'est-à-dire qui reste inchangée à une déformation Q-Gorenstein près. On classifie entièrement ces objets en trouvant 29 familles. On obtient ainsi un tableau contenant des modèles de ces surfaces, qui pour la plupart sont des intersections complètes dans des variétés toriques. Ce travail s'inscrit dans un contexte plus large, qui a pour cible de calculer leur cohomologie quantique pour ensuite vérifier si deux conjectures en symmetrie miroir. / This thesis concerns Fano varieties, which are central objects within the classification of algebraic varieties.The first problem we discuss involves smooth Fano varieties of dimension four. We study the potential singularities of an anticanonical divisor and determine their explicit local expression. As an intermediate step, we show that they are terminal points, that is the singularities which are closest to the smooth case from the point of view of birational geometry. We then show that the latter result generalizes in arbitrary dimension if we suppose that a nonvanishing conjecture of Kawamata holds.The second approach is to examine Fano varieties of smaller dimensions which admit singularities. The objects we consider are log del Pezzo surfaces with 1/3(1,1) points. This is the simplest example of a rigid singularity, that is it remains unchanged under Q-Gorenstein deformations. We give a complete classification of these surfaces, finding 29 families. We also provide a table describing almost all of them as complete intersections in toric varieties. This work belongs to an overarching project that aims at studying mirror symmetry for del Pezzo surfaces with cyclic quotient singularities.

In this thesis, we study the Hasse principle for curves and K3 surfaces. We give several sufficient conditions under which the Brauer-Manin obstruction is the only obstruction to the Hasse principle for curves and K3 surfaces. Using these sufficient conditions, we construct several infinite families of curves and K3 surfaces such that these families are counterexamples to the Hasse principle that are explained by the Brauer-Manin obstruction.

For a Fano variety V with at most Kawamata log terminal (klt) singularities and a finite group G acting bi-regularly on V , we say that V is G-exceptional (resp., G-weakly-exceptional) if the log pair (V,∆) is klt (resp., log canonical) for all G-invariant effective Q-divisors ∆ numerically equivalent to the anti-canonical divisor of V. Such G-exceptional klt Fano varieties V are conjectured to lie in finitely many families by Shokurov ([Sho00, Pro01]). The only cases for which the conjecture is known to hold true are when the dimension of V is one, two, or V is isomorphic to n-dimensional projective space for some n. For the latter, it can be shown that G must be primitive—which implies, in particular, that there exist only finitely many such G (up to conjugation) by a theorem of Jordan ([Pro00]). Smooth G-weakly-exceptional Fano varieties play an important role in non-rationality problems in birational geometry. From the work of Demailly (see [CS08, Appendix A]) it follows that Tian’s αG-invariant for such varieties is no smaller than one, and by a theorem of Tian such varieties admit G-invariant Kähler-Einstein metrics. Moreover, for a smooth G-exceptional Fano variety and given any G-invariant Kähler formin the first Chern class, the Kähler-Ricci iteration converges exponentially fast to the Kähler form associated to a Kähler- Einsteinmetric in the C∞(V)-topology. The termexceptional is inherited from singularity theory, to which this study enjoys strong links. We classify two-dimensional smooth G-exceptional Fano varieties (del Pezzo surfaces) and provide a partial list of all G-exceptional and G-weakly-exceptional pairs (S,G), where S is a smooth del Pezzo surface and G is a finite group of automorphisms of S. Our classification confirms many conjectures on two-dimensional smooth exceptional Fano varieties.

The goal of this thesis is to apply an approach due to Elkies to study the distribution of rational point counts for certain families of curves and surfaces over finite fields. A vector space of polynomials over a fixed finite field gives rise to a linear code, and the weight enumerator of this code gives information about point count distributions. The MacWilliams theorem gives a relation between the weight enumerator of a linear code and the weight enumerator of its dual code. For certain codes C coming from families of varieties where it is not known how to determine the distribution of point counts directly, we analyze low-weight codewords of the dual code and apply the MacWilliams theorem and its generalizations to gain information about the weight enumerator of C. These low-weight dual codes can be described in terms of point sets that fail to impose independent conditions on this family of varieties. Our main results concern rational point count distributions for del Pezzo surfaces of degree 2, and for certain families of genus 1 curves. These weight enumerators have interesting geometric and coding theoretic applications for small q. / Mathematics

Les courbes algébriques sont des objets centraux de la géométrie algébrique. Dans cette thèse, nous étudions ces objets sous différents angles de la géométrie algébrique tels que la géométrie algébrique effective et la géométrie arithmétique. Dans le premier chapitre, nous étudions les courbes non-hyperelliptiques de genre g et leurs jacobiennes liées par l’intermédiaire de diviseurs thêta caractéristiques. Ces derniers contiennent des propriétés géométriques extrinsèques qui permettent de calculer les constantes thêta. Dans le deuxième chapitre, nous nous concentrons sur les courbes hyperelliptiques de genre 2 et leur surface de Kummer associée avec une motivation cryptographique. Dans le troisième et dernier chapitre, nous étudions les revêtements doubles non-ramifiés des courbes non-hyperelliptiques de genre g pour obtenir des informations sur le p-rang. / Algebraic curves are central objects in algebraic geometry. In this thesis, we consider these objects from different angles of algebraic geometry such as computational algebraic geometry and arithmetic geometry. In the first chapter, we study non-hyperelliptic curves of genus g and their Jacobians linked via theta characteristic divisors. Such divisors provide extrinsic geometric properties which allow us to compute theta constants. In the second chapter, we focus on hyperelliptic curves of genus 2 and the associated Kummer surface with a cryptographic motivation. In the third and final chapter, we examine unramified double covers of non-hyperelliptic curves of genus g to obtain information about p-rank.