Ana J. Reguera

Families of arcs on rational surface singularities

We consider the spaceH of arcs on a rational surface singularity (S, P), with the proalgebraic structure induced by the truncation maps. We introduce some sets of arcs by imposing valuative conditions and we prove that they are closed subsets ofH. This leads to give a sufficient condition in order to have an affirmative answer for the problem of Nash. We conclude the solution of the problem for the minimal surface singularities.

A curve selection lemma in spaces of arcs and the image of the Nash map

We prove a finiteness result in the space of arcs

Towards the singular locus of the space of arcs

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Poincaré series of resolutions of surface singularities

of a singular variety X , which is an extension of the stability result of Denef and Loeser ( Germs of arcs on singular algebraic varieties and motivic integration , Invent. Math. 135 (1999), 201–232). From this follows a curve selection lemma for generically stable subsets of

Exceptional divisors that are not uniruled belong to the image of the Nash map

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The Denef-Loeser series for toric surface singularities

. As a consequence, we extend to all dimensions the problem of wedges, proposed by Lejeune-Jalabert ( Arcs analytiques et resolution minimale des singularites des surfaces quasi-homogenes , Lecture Notes in Mathematics, vol. 777 (Springer, Berlin, 1980), 303–336), and we obtain that an affirmative answer to this problem is equivalent to the surjectivity of the Nash map. This implies, for instance, that the Nash map is bijective for sandwiched surface singularities.

Arcs and wedges on sandwiched surface singularities

We give a theory on the space of arcs

Combinatorial aspects of sequences of point blowing ups

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POLARITY WITH RESPECT TO A FOLIATION

of a singular variety

Image of the Nash map in terms of wedges

X