the Zariski tangent space

On a smooth surface in space, at each point there is a tangent plane — the best flat approximation. The Zariski tangent space is the purely algebraic version of that tangent plane, defined for any point of any variety using only the coordinate ring, with no calculus or limits. At a smooth point it recovers exactly the tangent plane you expect; at a singular point, like the crossing of two lines or the tip of a cusp, it is too big, and that excess is precisely the algebraic signature of the singularity.

Fix a point p on a variety X with local ring O_{X,p} and maximal ideal m of functions vanishing at p. The Zariski tangent space is defined as the dual vector space T_p X = (m / m^2)*, a vector space over the residue field k. The quotient m / m^2 is the cotangent space: m collects functions vanishing at p, and squaring out m^2 throws away second-order and higher terms, leaving exactly the linear part — the differentials. A tangent vector is then a derivation, a linear functional on functions obeying the Leibniz rule. Concretely, for X = V(f_1, ..., f_r) in A^n, the tangent space at p is the kernel of the Jacobian matrix (partial f_i / partial x_j) evaluated at p: the solution space of the linearized equations, exactly the classical tangent plane.

The Zariski tangent space is the tool that distinguishes smooth from singular without ever leaving algebra. The point p is smooth (nonsingular) precisely when dim T_p X equals the dimension of X; it is singular when the tangent space is strictly larger, which happens exactly where the Jacobian drops rank. At the node of two crossing lines the tangent space is 2-dimensional though the curve is 1-dimensional; at a cusp it is also too big. Caveat: dim T_p X is always at least dim X and can jump up only at the singular locus, but reading the geometry off the tangent space alone can mislead — a cusp and a node have the same tangent-space dimension yet are different singularities, distinguished by higher-order data the tangent space cannot see.