Curves drawn on a hypersurface germ. (Courbes tracées sur un germe d’hypersurface.) (French) Zbl 0743.14002

Lejeune-Jalabert, Monique

Am. J. Math. 112, No. 4, 525-568 (1990).

This paper is motivated by some of the problems that J. Nash proposed in “Arc structure of singularities” in the midsixties. In this unpublished preprint, Nash considered parametrized formal arcs or curves on an algebraic variety \(V\) whose origin is in the singular set of \(V\). First he observed that, for each \(i\geq 0\), the space of their \(i\)-jets is a constructible set (in fact, this is a direct consequence of Artin’s approximation theorem).
Here we get a constructive proof of this property when \(V\) is an hypersurface with an isolated singular point. Our method is algorithmic and provides a linear bound for Artin’s \(\beta\)-function associated to the system of equations to solve, envolving the Milnor number and the multiplicity of the singular point. (Since then, this result has been generalized by M. Hickel.) Even if the singularity of the hypersurface \(V\) is not reduced to a point, the algorithm associates to each curve on \(V\) as above, a decreasing sequence of integers. If \(V\) is a branch of a curve in \(\mathbb{C}^ 2\), this sequence is equivalent to that of its Puiseux characteristic pairs.

Reviewer: M.Lejeune-Jalabert (Grenoble)

MSC:

14B12	Local deformation theory, Artin approximation, etc.
14J17	Singularities of surfaces or higher-dimensional varieties

Keywords:

Artin’s approximation theorem; hypersurface with an isolated singular point; \(\beta\)-function; Milnor number; multiplicity of the singular point