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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.

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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