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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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Metadaten
Verfasserangaben:Lukas Kühne
URN (Permalink):urn:nbn:de:hbz:386-kluedo-39864
Betreuer:Mathias Schulze
Dokumentart:Bachelorarbeit
Sprache der Veröffentlichung:Englisch
Veröffentlichungsdatum (online):12.02.2015
Jahr der Veröffentlichung:2015
Veröffentlichende Institution:Technische Universität Kaiserslautern
Titel verleihende Institution:Technische Universität Kaiserslautern
Datum der Annahme der Abschlussarbeit:29.07.2014
Datum der Publikation (Server):12.02.2015
Seitenzahl:36
Fachbereiche / Organisatorische Einheiten:Fachbereich Mathematik
DDC-Sachgruppen:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Klassifikation (Mathematik):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]
Lizenz (Deutsch):Standard gemäß KLUEDO-Leitlinien vom 28.10.2014