Varieties with too many rational points

We investigate Fano varieties defined over a number field that contain subvarieties whose number of rational points of bounded height is comparable to the total number on the variety.Comment: 23 page

Classifying sections of del Pezzo fibrations, I

We develop a strategy to classify the components of the space of sections of a del Pezzo fibration over $\mathbb{P}^{1}$. In particular, we prove the Movable Bend and Break lemma for del Pezzo fibrations. Our approach is motivated by Geometric Manin's Conjecture and proves upper bounds on the associated counting function. We also give applications to enumerativity of Gromov-Witten invariants and to the study of the Abel-Jacobi map.Comment: We generalized our main results to del Pezzo fibrations with Gorenstein terminal singularities. 51 page

Integral points of bounded height via universal torsors

A conjecture of Manin’s relates the number of rational points of bounded height on Fano varieties with their geometric properties. Analogously to this conjecture on rational points, we study the distribution of integral points of bounded height on three varieties: on a smooth Fano threefold of Picard number 2 and type 30 in the Mori–Mukai classification, on a quartic del Pezzo surface with an A1- and an A3-singularity, and on a toric threefold. We determine asymptotic formulas and interpret the leading term geometrically. For the proofs, we parametrize integral points using universal torsors, and use analytic techniques to count integral points on the torsor. This seems to be the first application of the torsor method to integral points. The asymptotic formula for our toric variety contradicts a result by Chambert-Loir and Tschinkel. We describe an obstruction that explains this contradiction, and study its relation with some constants that appear in asymptotic formulas for the number of integral points of bounded height.Eine Vermutung von Manin stellt einen Bezug zwischen der Anzahl rationaler Punkte beschränkter Höhe auf Fano-Varietäten und geometrischen Eigenschaften her. Analog zu dieser Vermutung für rationale Punkte untersuchen wir die Verteilung ganzer Punkte beschränkter Höhe auf drei Varietäten: auf einer glatten dreidimensionalen Fano-Varietät von Picardrang 2 und Typ 30 in der Mori– Mukai-Klassifikation, auf einer quartischen del-Pezzo-Fläche mit A1- und A3- Singularität und auf einer dreidimensionalen torischen Varietät. Wir bestimmen asymptotische Formeln und interpretieren den führenden Term geometrisch. In den Beweisen parametrisieren wir die ganzen Punkte mit Hilfe universeller Torsore, und zählen ganze Punkte auf den universellen Torsoren mit analytischen Methoden. Dies scheint die erste Anwendung der Torsor-Methode zum Zählen ganzer Punkte zu sein. Die asymptotische Formel für die torische Varietät steht im Widerspruch zu einem Ergebnis von Chambert-Loir und Tschinkel. Wir beschreiben eine Obstruktion, die diesen Widerspruch erklärt und untersuchen ihren Zusammenhang mit einigen Konstanten, die ein Bestandteil asymptotischer Formeln für die Anzahl ganzer Punkte beschränkter Höhe sind

Explicit Methods in Number Theory

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes the Sato-Tate conjecure, Langlands programme, function fields, L-functions and many other topics

Komplexe Analysis

The main aim of this workshop was to discuss recent developments in several complex variables and complex geometry. The topics included: classification of higher dimensional varieties, mirror symmetry, hyperbolicity, Kähler geometry and classical geometric questions

Enumerating Flux Vacua with Enhanced Symmetries

We study properties of flux vacua in type IIB string theory in several simple but illustrative models. We initiate the study of the relative frequencies of vacua with vanishing superpotential W=0 and with certain discrete symmetries. For the models we investigate we also compute the overall rate of growth of the number of vacua as a function of the D3-brane charge associated to the fluxes, and the distribution of vacua on the moduli space. The latter two questions can also be addressed by the statistical theory developed by Ashok, Denef and Douglas, and our results are in good agreement with their predictions. Analysis of the first two questions requires methods which are more number-theoretic in nature. We develop some elementary techniques of this type, which are based on arithmetic properties of the periods of the compactification geometry at the points in moduli space where the flux vacua are located.Comment: 83 pages, 8 figures, harvmac. v2: references added, typos fixed, brief discussion of complex conjugation in sec. 5 adde

Explicit Methods in Number Theory

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes asymptotics for field extensions and class numbers, random matrices and L-functions, rational points on curves and higher-dimensional varieties, and aspects of lattice basis reduction