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Search: MSC category 32S22 ( Relations with arrangements of hyperplanes [See also 52C35] )

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1. CJM Online first

Wang, Zhenjian
On algebraic surfaces associated with line arrangements
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
Categories:32S22, 32S25, 14J17, 14J29, 14J70

2. CJM 2011 (vol 63 pp. 1038)

Cohen, D.; Denham, G.; Falk, M.; Varchenko, A.
Critical Points and Resonance of Hyperplane Arrangements
If $\Phi_\lambda$ is a master function corresponding to a hyperplane arrangement $\mathcal A$ and a collection of weights $\lambda$, we investigate the relationship between the critical set of $\Phi_\lambda$, the variety defined by the vanishing of the one-form $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, and the resonance of $\lambda$. For arrangements satisfying certain conditions, we show that if $\lambda$ is resonant in dimension $p$, then the critical set of $\Phi_\lambda$ has codimension at most $p$. These include all free arrangements and all rank $3$ arrangements.

Keywords:hyperplane arrangement, master function, resonant weights, critical set
Categories:32S22, 55N25, 52C35

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