complex conjugate branches that cut the real branch at the origin. Singular points of smooth mappings As the notion of singular points is a purely local property, the above definition can be extended to cover the wider class of smooth mappings (functions from to where all derivatives exist). Analysis of these singular points can be reduced to the algebraic variety case by considering the jets of the mapping. The th jet is the Taylor series of the mapping truncated at degree and deleting the constant term. Nodes In classical algebraic geometry, certain special singular points were also called nodes. A node is a singular point where the Hessian matrix is non-singular; this implies that the singular point has multiplicity two and the tangent cone is not singular outside its vertex. See also Milnor map Resolution of singularities Singular point of a curve Singularity theory Smooth scheme Zariski tangent space References Algebraic varieties Singularity theory