Giancarlo Urzua : Arrangements of curves and algebraic surfaces

Posted by , part of the Algebraic Geometry Seminar.

At

March 11, 2008, 3 p.m.

In

SEO 636

Abstract

I will show a strong relation between Chern and logarithmic Chern numbers of complex algebraic surfaces. For a given arrangement of curves, there exist smooth projective surfaces with Chern ratio arbitrarily close to the logarithmic Chern ratio of the arrangement. The method is a random p-th root cover which exploits a large scale behavior of Dedekind sums and negative-regular continued fractions. I will emphasize that the random hypothesis is necessary for this limit result. For a certain large class of arrangements, this construction controls the topological fundamental group of the new surfaces. I will show how to obtain simply connected surfaces of general type with high Chern ratio, coming from complex line arrangements. Their Chern ratio is less than 8/3, being this upper bound the best possible for lines in the complex projective plane.