Arrangements of Curves and Algebraic Surfaces

We show a close relation between Chern and logarithmic Chern numbers of complex algebraic surfaces. The method is a “random” p-th root cover which exploits a large scale behavior of Dedekind sums and continued fractions. We use this to construct smooth projective surfaces with Chern ratio arbitrarily close to the logarithmic Chern ratio of a given arrangement of curves. For certain arrangements, this method allows us to control the irregularity and/or the topological fundamental group of the new surfaces. For example, we show how to obtain simply connected projective surfaces, which come from the dual Hesse arrangement, with Chern ratio arbitrarily close to 8 3 .