The Milnor fiber boundary of an arrangement determines its combinatorics

2025/12/05 by Sigurðsson, Baldur, Viu-Sos, Juan
#32S22 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2512.05891

Abstract

The boundary of the Milnor fiber associated with a complex line arrangement is a three dimensional plumbed manifold, and it is a combinatorial invariant. We prove the reverse implication, which was conjectured Némethi and Szilárd. That is, this boundary of the Milnor fiber determines the combinatorics of the arrangement. Furthermore, we give an explicit method which constructs the poset associated with the arrangement, given a plumbing graph in normal form for the boundary.

Citations

• Configurations of points and topology of real line arrangements
• Milnor fibre boundary of a non-isolated surface singularity
• Topology and combinatorics of real line arrangements

Related

• Milnor fibrations and oriented matroids
• The homology groups of the Milnor fiber associated to a central arrangement of hyperplanes in \CC³
• On algebraic surfaces associated to line arrangements
• Monodromy of triple point line arrangements
• First homology groups of the Milnor fiber boundary for generic hyperplane arrangements in ℂ³