On the plane generic projection of a branch of a curve (Q1086307)

Fix a branch \(C\subset {\mathbb{A}}^ 3(K)\), \(char(K)=0\), \(K=\bar K\), and let C' be its general projection into a plane. Here the authors give a new proof of a characterization due to Casas of the branches C such that C and C' have the same desingularization multiplicity sequence. The corresponding problem in \({\mathbb{A}}^ n(K)\) was solved by \textit{A. Campillo} and \textit{J. Castellanos} in Algebraic Geometry, Proc. int. Conf., La Rábida/Spain 1981, Lect. Notes Math. 961, 22-31 (1982; Zbl 0497.14012) in terms of the Hamburger-Noether matrix. The authors give also new criteria when C is either monomial or embedded in a surface with good tangent cone.