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.
The power structure over the Grothendieck (semi)ring of complex quasi-projective varieties constructed by the authors is used to express the generating series of classes of Hilbert schemes of zero-dimensional subschemes on a smooth quasi-projective variety of dimension d as an exponent of that for the complex affine space A d . Specializations of this relation give formulae for generating series of such invariants of the Hilbert schemes of points as the Euler characteristic and the Hodge-Deligne polynomial.
Abstract. Using superisolated singularities we present examples and counterexamples to some of the most important conjectures regarding invariants of normal surface singularities. More precisely, we show that the "Seiberg-Witten invariant conjecture"(of Nicolaescu and the third author), the "Universal abelian cover conjecture" (of Neumann and Wahl) and the "Geometric genus conjecture" fail (at least at that generality in which they were formulated). Moreover, we also show that for Gorenstein singularities (even with integral homology sphere links) besides the geometric genus, the embedded dimension and the multiplicity (in particular, the Hilbert-Samuel function) also fail to be topological; and in general, the Artin cycle does not coincide with the maximal (ideal) cycle.
For a germ of a meromorphic function f = P Q , we offer notions of the monodromy operators at zero and at infinity. If the holomorphic functions P and Q are non-degenerated with respect to their Newton diagrams, we give an analogue of the formula of Varchenko for the zeta-functions of these monodromy operators.
We generalize the notions of the orbifold Euler characteristic and of the higher order orbifold Euler characteristics to spaces with actions of a compact Lie group. This is made using the integration with respect to the Euler characteristic instead of the summation over finite sets. We show that the equation for the generating series of the k-th order orbifold Euler characteristics of the Cartesian products of the space with the wreath products actions proved by H. Tamanoi for finite group actions and by C. Farsi and Ch. Seaton for compact Lie group actions with finite isotropy subgroups holds in this case as well.
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