Editing Singularity (mathematics) (section)

==Algebraic geometry and commutative algebra==
In [[algebraic geometry]], a [[singular point of an algebraic variety|singularity of an algebraic variety]] is a point of the variety where the [[tangent space]] may not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like [[cusp (singularity)|cusps]]. For example, the equation {{math|1= ''y''{{sup|2}} − ''x''{{sup|3}} = 0}} defines a curve that has a cusp at the origin {{math|1= ''x'' = ''y'' = 0}}. One could define the {{math|''x''}}-axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the {{math|''x''}}-axis is a "double tangent."

For [[affine variety|affine]] and [[projective variety|projective varieties]], the singularities are the points where the [[Jacobian matrix]] has a [[rank (linear algebra)|rank]] which is lower than at other points of the variety.

An equivalent definition in terms of [[commutative algebra]] may be given, which extends to [[abstract variety|abstract varieties]] and [[scheme (mathematics)|schemes]]: A point is ''singular'' if the [[Localization of a ring|local ring at this point]] is not a [[regular local ring]].