Singular variety

In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth.

For example, the plane algebraic curve (a cubic curve) of equation

which is plotted below, crosses itself at the origin (0,0) and the origin is thus a double point of the curve. It is singular because a single tangent may not be correctly defined there.

More generally a plane curve defined by an implicit equation

where F is a smooth function is said to be singular at a point if the Taylor series of F has order at least 2 at this point.

The reason for that 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.

In general for a hypersurface

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