Coxeter-Dynkin diagrams of some space curve singularities
The author studies some invariants of certain space curves, by relating these to somewhat simpler ones. More precisely, he considers a germ \(X \subset \mathbb C^3\) of reduced complex curve, where \({X}\) is the union of a plane curve \(C\) and a transversal line. This is called a wedge singularity. The most interesting case is that where \(C\) is neither smooth nor quadratic (i.e., analytically isomorphic to a plane curve with equation \(x^2-y^2=0\)). The main objects attached to \(X\) studied in this paper are the discriminant of \(X\) (i.e. the discriminant \(D \subset S\) of the morphism \(p:{\mathcal X} \to S\) defining the semiuniversal deformation of \(X\)) and the Coxeter-Dynkin diagram of \(X\) (a useful tool to investigate the monodromy of \(X\), i.e., the natural action of \((\Pi_1(S-D),s)\) on \(\Aut (H_1(p^{-1}(s),\mathbb Z))\), for a general point \(s \in S\)). The main tool of this paper is a deformation of \(X\) into a new curve germ \((Y,0) \subset \mathbb C^3\), which is a complete intersection. The deformation is obtained by restricting \(p\) to a suitable subspace of \(S\) and it is equisingular, in the sense that the Milnor number \(\mu\) and \(\delta\) invariant remain constant. He shows, among other facts the following ones: (a) \(\delta (X) = \delta (C) +1\) and \(\mu(X)=\mu(C)+1\), (b) the Coxeter-Dynkin diagram and vanishing cycles of \(X\) can be easily described in terms of those of \(C\) and (c) the discriminants of \(X\) and \(C\) are closely related. Calculations of these invariants are easier for \(Y\), because this is a complete intersection. In the construction of the deformation of \(X\) into \(Y\), as well as in many proofs, the author uses the explicit description of the ideal of our space curve \(X\) as that generated by two by two minors of a suitable two by three matrix, and the fact that the semiuniveral deformation is obtained by perturbing the coefficients of this matrix. In the final section, by using techniques of real Morse Theory, he explicitly calculates Coxeter-Dynkin diagrams for some wedge singularities \(X\) of multiplicity three.
scientific article; zbMATH DE number 2168083