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Plane curves of minimal degree with prescribed singularities

  • We prove that there exists a positive \(\alpha\) such thatfor any integer \(\mbox{$d\ge 3$}\) and any topological types \(\mbox{$S_1,\dots,S_n$}\) of plane curve singularities, satisfying \(\mbox{$\mu(S_1)+\dots+\mu(S_n)\le\alpha d^2$}\), there exists a reduced irreducible plane curve of degree \(d\) with exactly \(n\) singular points of types \(\mbox{$S_1,\dots,S_n$}\), respectively. This estimate is optimal with respect to theexponent of \(d\). In particular, we prove that for any topological type \(S\) there exists an irreducible polynomial of degree \(\mbox{$d\le 14\sqrt{\mu(S)}$}\) having a singular point of type \(S\).

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Author:Gert-Martin Greuel, Christoph Lossen, Eugenii Shustin
URN (permanent link):urn:nbn:de:hbz:386-kluedo-7268
Document Type:Article
Language of publication:English
Year of Completion:1998
Year of Publication:1998
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2000/04/03
Tag:Singularity theory
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):14-XX ALGEBRAIC GEOMETRY / 14Bxx Local theory / 14B05 Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011