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摘要翻译：
证明了代数曲面的Chern不变量与对数Chern不变量之间的一个强关系。对于给定的曲线排列，我们发现Chern比任意接近于由该排列定义的对数曲面的对数Chern比的非奇异射影曲面。我们的方法是基于随机p次根覆盖序列，利用Dedekind和和连分式长度的某种大规模行为。我们证明了随机性对于我们的渐近结果是必要的，提供了“随机性蕴涵最优”的另一个例子。作为复数的应用，我们构造了具有大Chern比的一般类型的非奇异单连通射影曲面。特别地，我们改进了这类曲面Chern比的Persson-Peters-Xiao记录。
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英文标题：
《Arrangements of curves and algebraic surfaces》
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作者：
Giancarlo Urzua
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Number Theory 数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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英文摘要：
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.
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PDF链接：
https://arxiv.org/pdf/0711.0765
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