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On Algebraic Surfaces Associated with Line Arrangements
Published online by Cambridge University Press: 07 January 2019
Abstract
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For a line arrangement 
${\mathcal{A}}$ in the complex projective plane 
$\mathbb{P}^{2}$, we investigate the compactification 
$\overline{F}$ in 
$\mathbb{P}^{3}$ of the affine Milnor fiber 
$F$ and its minimal resolution 
$\tilde{F}$. We compute the Chern numbers of 
$\tilde{F}$ in terms of the combinatorics of the line arrangement 
${\mathcal{A}}$. As applications of the computation of the Chern numbers, we show that the minimal resolution is never a quotient of a ball; in addition, we also prove that 
$\tilde{F}$ is of general type when the arrangement has only nodes or triple points as singularities. Finally, we compute all the Hodge numbers of some 
$\tilde{F}$ by using some knowledge about the Milnor fiber monodromy of the arrangement.
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