Efficiency and complexity of hyperplane arrangements

TL;DR

This paper investigates conditions under which the monodromy eigenspaces of hyperplane arrangements are computable, introducing measures of arrangement complexity and efficiency, and establishing bounds for these conditions based on combinatorial data.

Contribution

It introduces the concepts of m-efficiency and m-complexity to quantify arrangement complexity and provides bounds ensuring the computability of monodromy eigenspaces.

Findings

Sufficient conditions are often met if m-efficiency is at most 2.

The m-complexity bound for sufficient conditions is at most (m+1)/2.

Introduces new combinatorial measures to analyze hyperplane arrangements.

Abstract

For a projective hyperplane arrangement, we study sufficient conditions in terms of combinatorial data for ESV-calculability of the monodromy eigenspaces of the first Milnor fiber cohomology for eigenvalues of order $m>1$. This can be reduced to the line arrangement case by Artin's theorem. These sufficient conditions are often unsatisfied if efficiency or complexity of the combinatorics of arrangement is high. In order to measure these, we introduce the notions of $m$-efficiency and $m$-complexity for $m\ge 3$. The former is defined to be the number of points with multiplicity divisible by $m$ lying on one line in average. In many cases, one of the above sufficient conditions is satisfied if it is at most 2, although there are certain exceptional cases, especially when $m=3$. The $m$-complexity is defined to be the maximal number of edges containing one vertex of the associated $m$-graph. We can show that one of the sufficient condition holds if it is at most $(m+1)/2$.