Titel:
Regular models of Fermat curves and applications to Arakelov theory ; Reguläre Modelle von Fermat-Kurven und Anwendungen in der Arakelov-Theorie
Beteiligte:
Curilla, Christian
[VerfasserIn]
Erschienen:
Staats- und Universitätsbibliothek Hamburg Carl von Ossietzky, 2010-01-01
Anmerkungen:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Beschreibung:
In this work we analyze the arithmetic and geometry of the Fermat curves of squarefree exponent N. Furthermore we consider applications to Arakelov theory. The main results are the construction of the minimal regular model over the ring of integers of the N-th cyclotomic field, and upper bounds for the arithmetic self-intersection number of the hermitian line bundle which is given by the canonical bundle equipped with the Arakelov metric. Starting point of our computation of the minimal regular model is the model which is given by the Fermat equation as well. We use the method of blowing-up, and we especially focus on the appropriate choice of the centers. The components of the special fiber which appear during the blowing-up process will be divided into different types that will be considered separately. In each step we use several methods to find the singular loci and verify regularity and normality. The main result describes the configuration of the special fiber of the minimal regular model together with the number, genus, self-intersection number and the multiplicity of the components. With a combinatoric argument, we get the transversality of the intersections. An advantage of our explicit construction is that we can use it for further applications related to the model itself. For example, for certain points of the generic fiber we can determine which components of the special fiber will be intersected by the horizontal divisors obtained as the Zariski-closure of these points. We can use this in order to construct a canonical divisor of this model with specific properties which are important for later applications. For the computation of the upper bounds of the arithmetic self-intersection number of the hermitian line bundle, we can use one of Kühn's results that reduces the situation to the computation of certain finite self-intersection numbers. We can calculate these numbers with our model and obtain asymptotic upper bounds. ; In dieser Arbeit untersuchen wir die Arithmetik und Geometrie der ...