Algebraic Geometry Seminar
Giancarlo Urzua
University of Michigan
Arrangements of curves and algebraic surfaces
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.
Tuesday March 11, 2008 at 3:00 PM in SEO 636