Discriminant of a generic projection of a minimal normal surface singularity

  • Romain Bondil
  • Published 2003

Abstract

Let (S, 0) be a rational complex surface singularity with reduced fundamental cycle, also known as a minimal singularity. Using a fundamental result by M. Spivakovsky, we explain how to get a minimal resolution of the discriminant curve for a generic projection of (S, 0) onto (C, 0) from the resolution of (S, 0). The material in this Note is organized as follows : § 1 recalls the definitions of polar curves, discriminants and a remarkable property of transversality due to Briançon-Henry and Teissier (thm. 1.2). For minimal surface singularities, a theorem due to M. Spivakovsky describes the behavior of the generic polar curve (cf. § 2). We use this theorem in § 3 to prove two lemmas relating on the one side the resolution of the generic polar curve to the resolution of a minimal surface singularity, and on the other side, the polar curve and the discriminant. Gathering these results, we give our main theorem in § 4, which provides us with a combinatorial way to describe the discriminant. 1 Polar curves and discriminants Let (S, 0) be a normal complex surface singularity (S, 0), embedded in (C , 0): for any (N − 2)-dimensional vector subspace D of C , we consider a linear projection C → C with kernel D and denote by pD : (S, 0) → (C , 0), the restriction of this projection to (S, 0). Restricting ourselves to the D such that pD is finite, and considering a small representative S of the germ (S, 0), we define, as in [11] (2.2.2), the polar curve C1(D) of the germ (S, 0) relative to the direction D, as the closure in S of the critical locus of the restriction of pD to S \ {0}. As explained in loc. cit., it makes sense to say that for an open dense subset of the Grassmann manifold G(N − 2, N) of (N − 2)-planes in C , the space curve C1(D) are equisingular in term of strong simultaneous resolutions. Then we define the discriminant ∆pD as (the germ at 0 of) the reduced analytic curve of (C, 0) image of C1(D) by the finite morphism pD.

Cite this paper

@inproceedings{Bondil2003DiscriminantOA, title={Discriminant of a generic projection of a minimal normal surface singularity}, author={Romain Bondil}, year={2003} }