Arrangements of curves and algebraic surfaces

Urzúa, Giancarlo

doi:10.1090/S1056-3911-09-00520-7

Author: Giancarlo Urzúa
Journal: J. Algebraic Geom. 19 (2010), 335-365
DOI: https://doi.org/10.1090/S1056-3911-09-00520-7
Published electronically: June 18, 2009
MathSciNet review: 2580678
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Abstract | References | Additional Information

Abstract: 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 $\mathbb {C}$, 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.

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Additional Information

Giancarlo Urzúa
Affiliation: Department of Mathematics and Statistics, University of Massachusetts at Amherst, 710 N. Pleasant Street, Amherst, Massachusetts 01003
MR Author ID: 797224
Email: urzua@math.umass.edu

Received by editor(s): November 22, 2007
Received by editor(s) in revised form: May 28, 2008
Published electronically: June 18, 2009
Additional Notes: The author was supported by a Fulbright-CONICYT Fellowship, and a Graduate Fellowship from the Department of Mathematics of the University of Michigan.
Dedicated: Dedicated to F. Hirzebruch on the occasion of his 80th birthday.