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On algebraic surfaces associated with line arrangements. (English) Zbl 1416.32013

Summary: 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.

MSC:

32S22 Relations with arrangements of hyperplanes
32S25 Complex surface and hypersurface singularities
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