On invariants of generic slices of weighted homogeneous corank 1 map germs from the plane to 3-space
Given a finitely determined corank 1 map germ \(f: (\mathbb{C}^2,0) \to (\mathbb{C}^3,0)\), one can recover information about the geometry of \(f\) by studying a generic transverse slice of the image of \(f\). In particular, in [Q. J. Math. 65, No. 4, 1375-1395 (2014; Zbl 1305.32015)], \textit{W. L. Marar} and \textit{J. J. Nuño-Ballesteros} considered \(C\), \(T\) and \(J\), the number of cusps, triple points and tacnodes of a stabilization of the transverse slice, which are related to the number of cross-caps \(C(f)\), triple points \(T(f)\) and both the delta invariant and \(N(f)\) of \(f\), respectively. In that paper, the author gives a formula for \(J\) in terms of the invariants of \(f\) and \(m(f(D(f)))\), the multiplicity of the image of the double point curve. The author gives a formula in terms of degrees and weights for \(m(f(D(f)))\) in the case that \(f\) is quasi-homogeneous. Using the Marar-Nuño-Ballesteros formula and the Mond expressions for the invariants of \(f\) in terms of degrees and weights, an expression is obtained for \(J\), too. Similar expressions were obtained in [Math. Proc. Camb. Philos. Soc. 166, No. 2, 353-369 (2019; Zbl 1421.32035)] by \textit{M. A. S. Ruas} and the author for the homogeneous case. Finally, the invariants \(J\) and \(m(f(D(f)))\) are calculated for all the simple germs from \((\mathbb{C}^2,0)\) to \((\mathbb{C}^3,0)\) in Mond's list.
scientific article; zbMATH DE number 7375734
scientific article