Smooth locus of a variety

The smooth locus consists of points $P$ for which $\dim T_{X,P}=\dim X$. For an irreducible variety over a perfect field it is a nonempty Zariski-open subset and therefore dense.

The closed subset of points where an algebraic variety is not smooth. Over a perfect field it is equivalently the locus whose local rings are not regular local rings. A normal variety has no singular points in codimension zero or one.

Over a perfect field, a point of a variety is smooth when its local ring is a regular local ring. For an irreducible variety of dimension $n$ at a closed point, this is equivalent to its Zariski tangent space having dimension $n$. The Jacobian criterion expresses the equality as a matrix-rank condition.

A point $p$ of an irreducible variety $X$ is singular when $\dim T_{p}X>\dim X$. Equivalently, its local ring is not regular. The singular locus is the complement of the smooth locus.

A point on $F=0$ 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 $2$ and factors into two distinct linear factors over an algebraic closure. The two factors give distinct tangent directions. In characteristic not $2$, $x^2+y^2+\text{higher terms}$ is an example.

If $X\subseteq {\mathbb{A}}^n$ is irreducible of dimension $d$, choose a nonzero $(n-d)$-rowed minor of a Jacobian matrix at one point of minimum tangent dimension. Its nonvanishing locus is a nonempty Zariski-open set on which the Jacobian has rank $n-d$, so every point there is smooth. Irreducibility makes every nonempty Zariski-open set dense.

For an affine variety over a perfect field, the tangent-space codimension at a point is the rank of the Jacobian matrix of defining equations. The smooth locus of a pure $d$-dimensional variety in ${\mathbb{A}}^N$ is where this rank is $N-d$.