Tangent spaces and Jacobian criteria

Let X be a reduced affine algebraic set over an algebraically closed field and p a point of X. A tangent vector v is an ambient vector for which every defining polynomial vanishes at p+εv modulo ε². Expanding a polynomial leaves its value at p plus ε times the directional derivative. The constant term already vanishes; setting each linear coefficient to zero gives a vector space. The dual-number symbol ε is formal and nilpotent, so this definition needs no topology, numerical step size, or analytic convergence.

If f₁ through f_r generate the full ideal I(X), their Jacobian rows suffice. For a combination h_i f_i, differentiation at p produces h_i(p)df_i(p), because f_i(p)=0 removes the other product-rule term. Thus the common kernel is unchanged by adding redundant ideal generators. This reasoning also works for an explicitly chosen equation scheme, but then the ideal must not be silently replaced by its radical. A squared equation may lose a first-order row while retaining nonzero nilpotent structure.

The tangent dimension is at least the local dimension at p. Nonsingularity means equality, or equivalently regularity of the local ring. For an irreducible d-dimensional affine variety over the algebraically closed field, every closed point has local dimension d, so the Jacobian rank criterion becomes rank n−d. For a reducible set, use the local dimension, the largest dimension among components through p. Merely subtracting the number of displayed equations ignores dependencies and can misclassify both smooth and singular points.

Over a perfect field, a finite-type variety is smooth at a point exactly when it is geometrically regular there; for perfect fields regularity and smoothness agree. Algebraically closed fields are perfect in every characteristic, so our criterion is not restricted to characteristic zero. Over an imperfect field, regularity alone is insufficient. For instance the field k[z]/(z^p−s), with k=F_p(s), is a regular zero-dimensional ring, yet its inseparable extension becomes nonreduced after adjoining a pth root of s. Its vanishing derivative signals failure of smoothness, not failure of its field ring to be regular.

The reduced point V(x) in the affine line has zero tangent dimension. The equation scheme defined by x² has the same ordinary point but a one-dimensional tangent space, because its equation has no linear term at zero. Neither computation is erroneous when its object is stated correctly. The mistake is claiming the tangent space of the reduced point while linearizing only the nonradical ideal. In symbolic work, establishing radicality or identifying the intended scheme is therefore part of the setup rather than an optional cleanup step.

The tangent space also differs from the tangent cone. At a cusp, the full linearized space can be an entire plane, while the lowest-degree nonzero equation singles out a repeated line in the tangent cone. A crossing likewise has the full plane as its tangent space but two lines in its tangent cone. The vector-space dimension diagnoses first-order excess; the cone retains leading higher-order directions and multiplicities. Neither should be described as the set of velocities of a single chosen parameterization without checking how that parameterization behaves at the singularity.