On Invariants of Generic Slices of Weighted Homogeneous Corank 1 Map Germs from the Plane to 3-Space

Abstract

In this work, we consider a quasi-homogeneous, corank 1, finitely determined map germ f from (C 2 , 0) to (C 3 , 0). We consider the invariants m(f (D(f))) and J , where m(f (D(f))) denotes the multiplicity of the image of the double point curve D(f) of f and J denotes the number of tacnodes that appears in a stabilization of the transversal slice curve of f (C 2). We present formulas to calculate m(f (D(f))) and J in terms of the weights and degrees of f .

On the topology of the transversal slice of a quasi-homogeneous map germ

“…Since is a germ of smooth curve in , . Now the proof follows by [ 13 , lemma 5·2] and [ 26 , theorem 3·2].…”

“… We note that if F adds some term of degree strictly greater than the degrees of f then it is topologically trivial but it may be not Whitney equisingular (see [ 24 , 5.5]). If g has corank 1 (and it is not necessarily quasi-homogeneous), Marar and Nuño–Ballesteros present in [ 12 , corollary 4·7] a characterization of Whitney equisingularity of F in terms of the constancy of the invariants C, T and J along the parameter space. If g is quasi-homogeneous, Mond shows in [ 18 ] (for any corank) that the invariants C and T are determined by the weights and degrees of g. The author shows in [ 26 , theorem 3·2] that the invariant J is also determined by the weights and degrees of f. Hence, it gives another proof of Corollary 4·4. …”

On the topology of the transversal slice of a quasi-homogeneous map germ

“…4.7] a characterization of Whitney equisingularity of F in terms of the constancy of the invariants C, T and J along the parameter space. If g is quasi-homogeneous, Mond shows in [19] (for any corank) that the invariants C and T are determined by the weights and degrees of g. The author shows in [25,Th. 3.2] that the invariant J is also determined by the weights and degrees of f .…”