On the trees associated to plane curve singularities and ideals
A possible approach to study the complexity of singularity of a reduced plane curve, or that of an ideal in a regular two dimensional local ring (primary to the maximal ideal) is to consider its associated tree of ``infinitely near points''. Each vertex of the tree corresponds to a point infinitely near to the singularity (or the support of the ideal), two vertices being joined by an edge if one is obtained from the other by means of a single quadratic transformation. The tree itself does not say much about the singularity (or ideal), but one may enrich it by assigning integral ``weights'' to the vertices. One obtains different notions of equivalence, or equisingularity, for curves (or ideals) by requiring that the various weighted trees that appear be isomorphic (by means of an isomorphism respecting the weights). However, the situation is different when we are dealing with curves which are members of the same family. In this case, several different natural notions of equivalence agree. More precisely, in this note we discuss, along the lines just mentioned, what we call the notions of \(R\)- and \(C\)-equivalence of ideals. The concept of \(R\)-equivalence was studied by \textit{J.-J. Risler} in Bull. Soc. Math. Fr. 101, 3-16 (1973; Zbl 0256.14006) and by \textit{A. Nobile} in Pac. J. Math. 170, No. 2, 543-566 (1995; Zbl 0903.14002). Also, there are discussed the notions of \(Z\)- and \(M\)-equivalence of reduced curves [\(Z\)-equivalence being the same as the well-known concept of Zariski-equivalence; see \textit{O. Zariski}, Am. J. Math. 87, 507-536 (1965; Zbl 0132.41601)]. In section 3 (theorem 2), it is checked that given a family of curves \(\{C_t\}\), \(t\) in a connected parameter space \(T\), then the different fibers are \(Z\)-equivalent to each other if and only if they are \(M\)-equivalent (although in general only one implication is valid). Concerning families of ideals, to require that all members be \(C\)-equivalent is strictly stronger that requiring \(R\)-equivalence, and the former condition (but not the second) is equivalent to requiring (essentially) that, ``in general'', an element of the defining ideal determines an equisingular family of plane curve singularities.
scientific article; zbMATH DE number 1089786