Zariski tangent space ($T_{p}X$)

For $P\in X\subseteq {\mathbb{A}}^n$, the Zariski tangent space is the common kernel at $P$ of the differentials of all polynomials in $I(X)$.

For a smooth algebraic surface defined by a homogeneous polynomial $F$ in projective space of dimension three, its tangent plane at $P$ is the displayed plane. Every projective line on the surface through $P$ lies in this plane.

Three coordinate axes in affine three-space have coordinate ideal $(xy,xz,yz)$; their common point has embedding dimension three. Three distinct coplanar lines meeting at one point have embedding dimension two there. An isomorphism preserves the unique point lying on all three irreducible components and preserves the algebraic cotangent space, so these two configurations are not isomorphic despite identical component incidence.

At a closed point of a variety over an algebraically closed field, the cotangent space is ${\mathfrak{m}}_P/{\mathfrak{m}}_P^2$. The universal property of Kähler differentials identifies its dual vector space with $k$-derivations of the local ring into the residue field, hence with the Zariski tangent space. The Kähler differential sheaf has the same fibre because a derivation kills constants and products of two elements of the maximal ideal.

For an affine variety, its dimension is the minimum of $\dim T_{X,P}$ over its points. A point is smooth when its tangent dimension attains this minimum and singular otherwise.

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 unibranch singular point of an algebraic variety on a plane algebraic curve. The ordinary cusp has local equation $y^2=x^3$ and parametrization $(x,y)=(t^2,t^3)$; 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 $F=0$ 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.

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 $2$ 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 $N-d$.

Since $T_{(x,y)}(X\times Y)=T_{x}X\oplus T_{y}Y$, if $Y$ is smooth thenIn particular, dimensions add.

Forin characteristic other than two,The variety has dimension two. Its tangent dimension is two when both coordinate pairs are nonzero, three when exactly one pair vanishes, and four at the origin.

The union of the three coordinate axes in ${\mathbb{A}}^3$ and a union of three distinct lines through the origin in ${\mathbb{A}}^2$ are not isomorphic. Their origins are intrinsically the unique points common to all three irreducible components, but their Zariski tangent spaces there have dimensions three and two respectively.