• search hit 1 of 1
Back to Result List

Freeness of hyperplane arrangements with multiplicities

  • This bachelor thesis is concerned with arrangements of hyperplanes, that is, finite collections of hyperplanes in a finite-dimensional vector space. Such arrangements can be studied using methods from combinatorics, topology or algebraic geometry. Our focus lies on an algebraic object associated to an arrangement \(\mathcal{A}\), the module \(\mathcal{D(A)}\) of logarithmic derivations along \(\mathcal{A}\). It was introduced by K. Saito in the context of singularity theory, and intensively studied by Terao and others. If \(\mathcal{D(A)}\) admits a basis, the arrangement \(\mathcal{A}\) is called free. Ziegler generalized the concept of freeness to so-called multiarrangements, where each hyperplane carries a multiplicity. Terao conjectured that freeness of arrangements can be decided based on the combinatorics. We pursue the analogous question for multiarrangements in special cases. Firstly, we give a new proof of a result of Ziegler stating that generic multiarrangements are totally non-free, that is, non-free for any multiplicity. Our proof relies on the new concept of unbalanced multiplicities. Secondly, we consider freeness asymptotically for increasing multiplicity of a fixed hyperplane. We give an explicit bound for the multiplicity where the freeness property has stabilized.

Download full text files

Export metadata

Additional Services

Share in Twitter Search Google Scholar
Author:Lukas Kühne
URN (permanent link):urn:nbn:de:hbz:386-kluedo-39864
Advisor:Mathias Schulze
Document Type:Bachelor Thesis
Language of publication:English
Publication Date:2015/02/12
Year of Publication:2015
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2014/07/29
Date of the Publication (Server):2015/02/12
Number of page:36
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):14-XX ALGEBRAIC GEOMETRY / 14Nxx Projective and enumerative geometry [See also 51-XX] / 14N20 Configurations and arrangements of linear subspaces
16-XX ASSOCIATIVE RINGS AND ALGEBRAS (For the commutative case, see 13-XX) / 16Wxx Rings and algebras with additional structure / 16W25 Derivations, actions of Lie algebras
52-XX CONVEX AND DISCRETE GEOMETRY / 52Cxx Discrete geometry / 52C35 Arrangements of points, flats, hyperplanes [See also 32S22]
Licence (German):Standard gemäß KLUEDO-Leitlinien vom 28.10.2014