Pencil type line arrangements of low degree: classification and monodromy

TL;DR

This paper classifies certain low-degree pencil-type line arrangements, specifically (3,3)-nets and (3,4)-nets with only double and triple points, and explores how monodromy relates to their combinatorial structure.

Contribution

It provides a complete classification of specific low-degree nets and establishes a link between monodromy non-triviality and pencil-type arrangements for arrangements with up to 14 lines.

Findings

Exactly 3 effective arrangements for each net type up to lattice isomorphism.

Some arrangements are new examples of pencil-type line arrangements.

Monodromy non-triviality implies arrangements are of reduced pencil-type for arrangements with ≤14 lines.

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 H^1(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.