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 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.
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.