On rational cuspidal projective plane curves



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Melle Hernández, Alejandro and Fernández de Bobadilla de Olarzábal, Javier José and Luengo Velasco, Ignacio and Némethi , A. (2006) On rational cuspidal projective plane curves. Proceedings of the London Mathematical Society , 92 (1). pp. 99-138. ISSN 0024-6115

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In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

Item Type:Article
Uncontrolled Keywords:Singularities; Invariants; Surfaces; Monodromy; Number; Links
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:13927
Deposited On:21 Nov 2011 08:16
Last Modified:06 Feb 2014 09:55

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