Projective dimensions of hyperplane arrangements
Abstract
We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for projective dimensions of hyperplane arrangements. They are generalizations of the free arrangement cases, that can be regarded as the special case of our result when the projective dimension is zero. The keys to prove them are several new methods to determine the surjectivity of the Euler and the Ziegler restriction maps, that is combinatorial when the projective dimension is not maximal for all localizations. Also, we introduce a new class of arrangements in which the projective dimension is comibinatorially determined.
Cite
@article{arxiv.2009.04101,
title = {Projective dimensions of hyperplane arrangements},
author = {Takuro Abe},
journal= {arXiv preprint arXiv:2009.04101},
year = {2021}
}
Comments
44 pages. In version 3, the statements and proofs of Theorems 1.5, 1.6 are corrected. Also, the proofs of Theorems, 5.1 and 5.2 are revised