Singular point of an algebraic variety

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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 that is not singular is said to be regular. An algebraic variety that has no singular point is said to be non-singular or smooth.

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

${\displaystyle (x-x_{0})F'_{x}(x_{0},y_{0})+(y-y_{0})F'_{y}(x_{0},y_{0})=0,}$

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

${\displaystyle F(x,y,z,\ldots )=0}$

the singular points are those at which all the

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).^[1]

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

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

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