Topics in Hyperplane arrangements - Errata

TL;DR

This paper corrects a previously published formula related to hyperplane arrangements, clarifying the conditions under which it holds and updating related propositions to ensure mathematical accuracy.

Contribution

It provides a corrected version of a key proposition in the theory of hyperplane arrangements, clarifying the assumptions and fixing the formula in the existing literature.

Findings

Corrected Proposition 8.5 for hyperplane arrangements

Updated related Proposition 9.2 accordingly

Clarified the conditions for the formula's validity

Abstract

We have received an e-mail from Bryan Gillespie pointing out that a proposition, that is Proposition 8.5, of our book, [1] is incorrect as stated. The given formula (8.5) is valid only in the generic case that is assuming that for any point of the arrangement $p, X_p$ is formed by a basis. The correct proposition is slightly weaker, in general one must replace Formula 8.5 of the book with the next Formula (3). Accordingly one has to change Proposition 9.2 in the obvious way. The remaining parts of the book are not affected but one should remove the first line of 11.3.3 which quotes the incorrect formula. Here we discuss the correct proposition, replacing Proposition 8.5.