Arrangements of curves and algebraic surfaces

By Giancarlo Urzúa. 2007

63 citations recorded by the source corpus.

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 the complex numbers, 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.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6fb7d0be048073e0343785cf28d66d32311cbb05

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In Geometry and Topology

Citations in the corpus

Line arrangements and configurations of points with an unexpected geometric property (2016, methodology): “We use this result to study so-called Fermat arrangements of lines [U].”
Local Systems on Proper Algebraic V -manifolds (2010, methodology): “This covering technique has been used by many authors since then; for a recent example see Urzúa [106].”