Deletion theorem and combinatorics of hyperplane arrangements
Combinatorics
2017-09-26 v2 Commutative Algebra
Algebraic Geometry
Abstract
We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give an explicit sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials. This gives a lot of corollaries including the existence of free filtrations. The proof is based on the result about the form of minimal generators of a logarithmic derivation module of a multiarrangement which satisfies the -equality.
Cite
@article{arxiv.1709.05842,
title = {Deletion theorem and combinatorics of hyperplane arrangements},
author = {Takuro Abe},
journal= {arXiv preprint arXiv:1709.05842},
year = {2017}
}
Comments
15 pages. In Versiion 2, a negative answer to Problem 3.3 is given in Remark 3.4