Arcs and wedges on sandwiched surface singularities. (English) Zbl 0960.14015

Lejeune-Jalabert, Monique; Reguera-López, Ana J.

Am. J. Math. 121, No. 6, 1191-1213 (1999).

Let \((S,P)\) be a surface singularity at \(P\). An arc on \((S,P)\) is an algebroid curve going through \(P\) on \(S\), given by formal power series in one variable. An arc is general on \((S,P)\) if its strict transform on the minimal desingularization is smooth and intersects the exceptional curve \(E\) transversally at a point lying on a Zariski dense open set of regular points of \(E\). A wedge on \((S,P)\) is a parametrization of \(S\) by formal power series in two variables; the wedge is centered at an arc if its parametrization comes from the wedge parametrization by simple substitution of two one-variable series. Generally speaking the authors show that a wedge centered at a general arc on a normal surface singularity \((S,P)\) lift to its minimal desingularization provided \((S,P)\) is a sandwiched singularity, i.e. it is the formal neighborhood of \(P\) on \(S\) obtained by blowing up a complete ideal \(I\) in the local ring of a closed point \(O\) on a non-singular algebraic surface \({\mathcal X}_0\) defined over an algebraically closed field \(k\).

Reviewer: Stanislaw Janeczko (Warszawa)

MSC:

14J17	Singularities of surfaces or higher-dimensional varieties
13F25	Formal power series rings
14M05	Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14B05	Singularities in algebraic geometry

Keywords:

surface singularity; algebroid curve; wedge centered at a general arc; minimal desingularization; sandwiched singularity