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Concept
Singular point of an algebraic variety
Summary
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.
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 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.
In general for a hypersurface
the singular points are those at which all the partial derivatives simultaneously vanish. A general algebraic variety V being defined as the common zeros of several polynomials, the condition on a point P of V to be a singular point is that the Jacobian matrix of the first order partial derivatives of the polynomials has a rank at P that is lower than the rank at other points of the variety.
Points of V that are not singular are called non-singular or regular. It is always true that almost all points are non-singular, in the sense that the non-singular points form a set that is both open and dense in the variety (for the Zariski topology, as well as for the usual topology, in the case of varieties defined over the complex numbers).
In case of a real variety (that is the set of the points with real coordinates of a variety defined by polynomials with real coefficients), the variety is a manifold near every regular point.
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