Singular point of an algebraic variety

The reason for this is that, in differential calculus, the tangent at the point (x0, y0) of such a curve is defined by the equation whose left-hand side is the term of degree one of the Taylor expansion.

Thus, if this term is zero, the tangent may not be defined in the standard way, either because it does not exist or a special definition must be provided.

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 M to Rn 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 kth jet is the Taylor series of the mapping truncated at degree k and deleting the constant term.

The plane algebraic curve (a cubic curve ) of equation y 2 − x 2 ( x + 1) = 0 crosses itself at the origin (0, 0) . The origin is a double point of this curve. It is singular because a single tangent may not be correctly defined there.