ARRANGEMENTS OF CURVES AND ALGEBRAIC SURFACES

We prove a strong relation between Chern and log Chern invariants of algebraic surfaces. For a given arrangement of curves, we find nonsingular projective surfaces with Chern ratio arbitrarily close to the log Chern ratio of the log surface defined by the arrangement. Our method is based on sequences of random p-th root covers, which exploit a certain large scale behavior of Dedekind sums and lengths of continued fractions. We show that randomness is necessary for our asymptotic result, providing another instance of "randomness implies optimal". As an application over C, we construct nonsingular simply connected projective surfaces of general type with large Chern ratio. In particular, we improve the Persson-Peters-Xiao record for Chern ratios of such surfaces.