Ignacio Luengo Velasco

Classification of Rational Unicuspidal Projective Curves whose Singularities Have one Puiseux Pair

It is a very old and interesting open problem to characterize those collections of embedded topological types of local plane curve singularities which may appear as singularities of a projective plane curve C of degree d. The goal of the present article is to give a complete (topological) classification of those cases when C is rational and it has a unique singularity which is locally irreducible (i.e., C is unicuspidal) with one Puiseux pair.

ON RATIONAL CUSPIDAL PLANE CURVES, OPEN SURFACES AND LOCAL SINGULARITIES

Let C be an irreducible projective plane curve in the complex projective space P(2). The classification of such curves, up to the action of the automorphism group PGL(3, C) on P(2), is a very difficult open problem with many interesting connections. The main goal is to determine, for a given d, whether there exists a projective plane curve of degree d having a fixed number of singularities of given topological type. In this note we are mainly interested in the case when C is a rational curve. The aim of this article is to present some of the old conjectures and related problems, and to complete them with some results and new conjectures from the recent work of the authors.

On the power structure over the Grothendieck ring of varieties and its applications

We discuss the notion of a power structure over a ring and the geometric description of the power structure over the Grothendieck ring of complex quasi-projective varieties and show some examples of applications to generating series of classes of configuration spaces (for example, nested Hilbert schemes of J. Cheah) and wreath product orbifolds.

Superisolated Surface Singularities

In this survey, we review part of the theory of superisolated surface singularities (SIS) and its applications including some new and recent developments. The class of SIS singularities is, in some sense, the simplest class of germs of normal surface singularities. Namely, their tangent cones are reduced curves and the geometry and topology of the SIS singularities can be deduced from them. Thus this class contains, in a canonical way, all the complex projective plane curve theory, which gives a series of nice examples and counterexamples. They were introduced by I. Luengo to show the non-smoothness of the μ-constant stratum and have been used to answer negatively some other interesting open questions. We review them and the new results on normal surface singularities whose link are rational homology spheres. We also discuss some positive results which have been proved for SIS singularities

Germs of holomorphic vector fields in C3 without a separatrix

Oblatum 28-XII-1988 & 26-III-1990 & 10-X-1990 & 13-VI-1991 & 30-IX-1991. Supported by Universite de Toulouse, CONACYT-CNRS and CAICYT-87-336 Fulltext Preview

Spiders and multiplicity sequences

The spider principle is used for establishing a formula for a finite quadratic sequence which determines the multiplicity sequences of all the sprouts which are founded upon the given finite quadratic sequence. This formula is basic for the theories of curvettes and dicriticals.