On the Alexander polynomials of conic-line arrangements

2023/05/02 by Alexandru Dimca, Dimca, Alexandru, Piotr Pokora +3
Mathematics · #13D02 #14B05 #32S35 #32S40 #32S55 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Primary: 14H50 #Secondary: 14N20

paper · pdf · doi:10.48550/arxiv.2305.01450

openalex publication_date 2023/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

In the present paper we compute Alexander polynomials for certain classes of conic-line arrangements in the complex projective plane which are related to pencils. We prove two general results for curve arrangements coming from Halphen pencils of index k≥ 2. Then we apply them to the Hesse arrangement of conics and to some of its degenerations. The results are completed by computations using computer algebra. In particular, we construct conic-line arrangements which are non-reduced pencil-type arrangements and have as roots of their Alexander polynomials roots of unity of order 7. Such roots are not known and are conjectured not to exist in the class of line arrangements.

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