Homology graph of real arrangements and monodromy of Milnor fiber

2016/06/30 by Pauline Bailet, Simona Settepanella
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Biology #Combinatorics #Gene #Genetics #Graph #Homology (biology) #Mathematics #Monodromy #Pure mathematics #math.AT #msc:05B35 #msc:14-XX #msc:20F36 #msc:52B35 #msc:52C35

paper · pdf · doi:10.1016/j.aam.2017.04.006

published as Advances in Applied Mathematics, 20 (2017) 46-85 · 35 pages, 15 figures

arxiv created 2017/03/30 · openalex publication_date 2017/05/05 · arxiv updated 2017/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study the first homology group of the Milnor fiber of sharp arrangements in the real projective plane. Our work relies on the minimal Salvetti complex of the deconing arrangement and its boundary map. We describe an algorithm which computes possible eigenvalues of the first monodromy operator. We prove that, if a condition on some intersection points of lines is satisfied, then the only possible non trivial eigenvalues are cubic roots of the unity. Moreover we give sufficient conditions for just eigenvalues of order 3 or 4 to appear in cases in which this condition is not satisfied.

Citations

• Arrangements of Hyperplanes
• Monodromy of triple point line arrangements
• A catalogue of simplicial arrangements in the real projective plane
• Minimality of hyperplane arrangements and basis of local system cohomology
• The chamber basis of the Orlik–Solomon algebra and Aomoto complex
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