Seshadri-Kostanten auf abelschen Flächen

Christoph Schulz, Seshadri-Konstanten auf abelschen Flächen, Zusammenfassung: Gegenstand dieser Arbeit sind Seshadri-Konstanten von amplen Geradenbündeln auf glatten projektiven komplexen Varietäten. Zu einem amplen Geradenbündel L und einem Punkt x definiert man die Seshadri-Konstante dur...

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Bibliographic Details
Main Author: Schulz, Christoph
Contributors: Bauer, Thomas (Prof. Dr.) (Thesis advisor)
Format: Doctoral Thesis
Published: Philipps-Universität Marburg 2004
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Christoph Schulz, Seshadri constants on abelian surfaces, Abstract: The main subject of this thesis are Seshadri constants of ample line bundles on smooth projective complex varieties. For an ample line bundle L and a point x one defines the Seshadri constant to be the supremum of all positive numbers e, such that the bundle f*L-eE is still nef over the blow up of the variety in the point x with exceptional divisor E. With this one measures the 'local' positivity of the line bundle L in the point x. It shows that Seshadri constants are difficult to calculate explicitly, even with using specific geometric properties of the variety. The aim will be to find methods for the calculation of Seshadri constants on abelian surfaces. After showing some methods for general projective complex surfaces in chapter 1, we focus on special methods for abelian surfaces in chapter 2 and 3. We will present a collection of known and new ideas, where the central new results are given in chapter 3. Here we calculate Seshadri constants on products of elliptic curves: (1) the product of two elliptic curves, which are not isogenous, (2) the product of an elliptic curve without complex multiplication with itself, (3) the product of special elliptic curves with complex multiplication with itself. We succeed to calculate the Seshadri constants for all given ample line bundles in these cases. Doing this, we are able to give the first known results for abelian surfaces with Picard number 3 and 4. It shows that the elliptic curves on these surfaces play a important role for the calculation of the Seshadri constants. Therefore we give a parameterisation of the numerical equivalence classes of all elliptic curves on these surfaces. Having this at hand, we use methods from the geometry of numbers to find submaximal curves. In this way we can give explicit formulas for the calculation of the Seshadri constants, using the numerical equivalence class of the line bundle, and examine the behaviour of the Seshadri function on the nef cone. After an introduction the basic notions, like Seshadri constants, submaximality and abelian surfaces are explained in chapter one. The second chapter deals with Seshadri constants of abelian surfaces. After giving some general results and bounds, we show the first calculations for simple abelian surfaces, particularly the result of Thomas Bauer for simple abelian surfaces with Picard number 1. The third chapter is devoted to the calculation of Seshadri constants on not simple abelian surfaces. Especially products of two elliptic curves with and without complex multiplication are studied in detail.