Pencil type line arrangements of low degree: classification and monodromy

2013/05/22 by Alexandru Dimca, Dimca, Alexandru, Denis Ibadula +4
Mathematics · #14C21 Secondary 58K10 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Primary 52C35 #math.AG #math.AT #msc:14C21 #msc:52C35 #msc:58K10

paper · pdf · doi:10.48550/arxiv.1305.5092

Remark 2.4, (new) Example 2.3 and a couple of references added, typo corrected in Example 2.6 M_1(1) and M_1(2)

openalex publication_date 2013/05/22 · arxiv created 2013/07/25 · arxiv updated 2013/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The complete classification of (3,3)-nets and of (3,4)-nets with only double and triple points is given. Up to lattice isomorphism, there are exactly 3 effective possibilities in each case, and some of these provide new examples of pencil-type line arrangements. For arrangements consisting of at most 14 lines and having points of multiplicity at most 5, we show that the non-triviality of the monodromy on the first cohomology H¹(F) of the associated Milnor fiber F implies the arrangement is of reduced pencil-type. In particular, the monodromy is determined by the combinatorics in such cases.

Citations

• Moduli spaces of ten-line arrangements with double and triple points
• Realization of finite Abelian groups by nets in P²
• Higher Homotopy Groups of Complements of Complex Hyperplane Arrangements

Related

• Monodromy of triple point line arrangements
• On two pencils of cubic curves
• On line arrangements with applications to 3-nets
• Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements
• The Milnor fiber boundary of an arrangement determines its combinatorics