A point of an irreducible variety is singular when . Equivalently, its local ring is not regular. The singular locus is the complement of the smooth locus.
A unibranch singular point of an algebraic variety on a plane algebraic curve. The ordinary cusp has local equation and parametrization ; its branch has a repeated tangent and a vanishing first derivative at the singular point. Higher-order cusps may have different local equations. A cuspidal cubic gives the basic projective example.
A point on where all first partial derivatives vanish. For a reduced hypersurface this is the Jacobian criterion; a nondegenerate quadratic leading part defines an ordinary double point.
Restrict a cubic equation to the affine line through two distinct singular points. At each point the equation and its first directional derivative vanish, so the restricted polynomial has two distinct double roots. A polynomial of degree at most three with this property is identically zero. Hence the whole line lies in the hypersurface. The multiplicity argument works in every characteristic.
For a reduced plane algebraic curve, an ordinary double point is a point whose lowest nonzero local homogeneous term has degree and factors into two distinct linear factors over an algebraic closure. The two factors give distinct tangent directions. In characteristic not , is an example.
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A singular point of an algebraic variety is a point where the variety is not well-behaved in terms of its geometric structure. More formally, a point \( P \) on an algebraic variety \( V \) defined by a set of polynomial equations is termed a singular point if the local behavior of the variety at that point exhibits some form of "singularity," meaning that it fails to meet certain smoothness conditions.