Monodromy of triple point line arrangements

TL;DR

This paper demonstrates that for certain line arrangements with limited complexity, the monodromy operator's action on the Milnor fiber's cohomology is determined by combinatorial data, with a possible exception.

Contribution

It establishes a combinatorial determination of monodromy for line arrangements with up to 18 lines and at most triple points, extending understanding of monodromy in algebraic geometry.

Findings

Monodromy action is combinatorially determined for arrangements with ≤18 lines.

Possible exception identified for arrangements exceeding certain complexity.

Provides new insights into the relationship between combinatorics and topology of line arrangements.

Abstract

We show that the monodromy operator action on the first cohomology group of the Milnor fiber is combinatorially determined for line arrangements with at most triple points and containing at most 18 lines, with one possible exception.