On two pencils of cubic curves

TL;DR

This paper presents examples of complex cubic curve arrangements with monodromy eigenvalues of order 5 and 6, revealing unexpected behaviors in curve arrangements close to line arrangements.

Contribution

It introduces specific cubic curve arrangements exhibiting non-trivial monodromy eigenvalues, expanding understanding beyond line arrangements.

Findings

Monodromy eigenvalues of order 5 and 6 in cubic arrangements

Examples close to line arrangements with complex monodromy behavior

Use of Dimca and Sticlaru's algorithm for detection

Abstract

We give two examples of plane curve arrangements of pencil type which are very close to line arrangements, though the action of the monodromy operator on the first cohomology group of the Milnor Fiber has eigenvalues of order 5 and 6, showing that surprising situations can occur for larger classes of curve arrangements than for line arrangements. Our computations rely on the algorithm given by A. Dimca and G. Sticlaru which detects the non trivial monodromy eigenspaces of free curves.