Cokernels of the Euler restriction map of logarithmic derivation modules

TL;DR

This paper investigates the cokernel of the Euler restriction map in logarithmic derivation modules of plane arrangements, providing bounds and formulas related to the arrangement's combinatorics and freeness.

Contribution

It introduces the first study of the cokernel of the Euler restriction map, offering bounds and explicit formulas in the free arrangement case.

Findings

Upper bound for the cokernel dimension

Formula for the cokernel when the arrangement is free

Connection to the characteristic polynomial

Abstract

There are two restriction maps of the logarithmic modules of plane arrangements in a three dimensional vector space. One is the Euler restriction and the other is the Ziegler restriction. The dimension of the cokernel of the Ziegler restriction map of logarithmic derivation modules has been well-studied for the freeness of hyperplane arrangements after Yoshinaga's celebrated criterion for freeness, which connects the second Betti number and the splitting type (exponents). However, though the Euler restriction has a longer history than the Ziegler restriction, the cokernel and its dimension of the Euler restriction have not been studied at all. The aim of this article is to study the cokernel and dimension of the Euler restriction maps in terms of combinatorics, more explicitly, the characteristic polynomial. We give an upper bound of that cokernel, and show the formula for that if the arrangement is free.