Arc structure of singularities. (English) Zbl 0880.14010

Nash, John F. jun.

Duke Math. J. 81, No. 1, 31-38 (1995).

The author introduces a new approach in the study of singularities of algebraic varieties, the investigation of its “arc structure”. In his terminology, an arc on a variety \(V\) defined over the complex numbers, is a germ of a parametrized algebroid curve on the variety, i.e. a morphism from \((T,0)= \text{Spec} \mathbb{C} [[t]]\) into \(V\), given by convergent power series. Finiteness properties of the set of arcs on \(V\), which do not factor through the singular locus, and send 0 into a given algebraic subvariety \(W\) of \(V\), are derived from the existence of a resolution of singularities of \(V\). A decomposition of this set into a finite number of families, depending only on the pair \((V,W)\), follows naturally; moreover for any resolution \(X \) of \(V\), the families are shown to be in \(1-1\) correspondence with a subset of the irreducible components of the inverse image of \(W\) in \(X\). The components so distinguished are essential components with respect to \(W\), appearing up to birational equivalence in all resolutions. A natural question arises; does every essential component come from an arc family? For surfaces, this amounts to decide whether there are as many families of arcs associated with \(W\) as irreducible components of its inverse image in the minimal resolution. A few examples are presented at the end.
The paper has been written in the midsixties. D. Mumford mentions its influence in the introduction of the book by G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, “Toroidal embeddings” I, Lect. Notes Math. 339 (1973; Zbl 0271.14017)]. The problems stated above remain unsolved at the present time. Reference to further related development are given in “Sur l’espace des courbes tracées sur une singularité” by G. Gonzalez-Sprinberg and M. Lejeune-Jalabert [in: Algebraic Geometry and Singularities, Proc. 3rd internat. Conf., La Rabida 1991, Prog. Math. 134, 9-32 (1996; Zbl 0862.14003)].

Reviewer: M.Lejeune-Jalabert

MSC:

14E15	Global theory and resolution of singularities (algebro-geometric aspects)
14H20	Singularities of curves, local rings

Keywords:

resolution of singularities; arc structure; singularities; essential components

Citations:

Zbl 0271.14017; Zbl 0862.14003

References:

[1]	C. Chevalley, Fondements de la géométrie algébrique , Secrétariat Mathématique, Paris, 1958. · Zbl 0087.35501
[2]	H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II , Ann. of Math. (2) 79 (1964), 109-203; ibid. (2) 79 (1964), 205-326. JSTOR: · Zbl 0122.38603 · doi:10.2307/1970486