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# On algebraic surfaces associated with line arrangements

Published:2018-04-04

• Zhenjian Wang,
CNRS, LJAD, UMR 7351, Univ. Nice Sophia Antipolis , 06100 Nice , France
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## Abstract

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 $\widetilde{F}$. We compute the Chern numbers of $\widetilde{F}$ 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 $\widetilde{F}$ is of general type when the arrangement has only nodes or triple points as singularities; finally, we compute all the Hodge numbers of some $\widetilde{F}$ by using some knowledge about the Milnor fiber monodromy of the arrangement.
 Keywords: line arrangement, Milnor fiber, algebraic surface, Chern number
 MSC Classifications: 32S22 - Relations with arrangements of hyperplanes [See also 52C35] 32S25 - Surface and hypersurface singularities [See also 14J17] 14J17 - Singularities [See also 14B05, 14E15] 14J29 - Surfaces of general type 14J70 - Hypersurfaces

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