(Zariski tangent space)
For \(p \in X\) with local ring \(\OO_{X,p}\) and maximal ideal \(\mfm_p\), the Zariski tangent space is \[ T_p X \da \dualof{(\mfm_p / \mfm_p^2)} . \] The point \(p\) is smooth if \(\dim_k T_p X = \dim_p X\), and singular otherwise.
(Jacobian criterion)
Let \(X\subseteq\AA^n\) be an affine variety of dimension \(d\), let \(I(X) = \gens{f_1,\ldots,f_t} \subseteq k[x_1,\ldots,x_n]\), and let \(J_X(p) = \left[ \partial f_i / \partial x_j (p) \right]\). Then \(\rank J_X(p)\le n-d\) for every \(p\in X\), and \(p\) is a smooth point if and only if \(\rank J_X(p) = n-d\) [@Har10a, Theorem I.5.1].
For every \(p\in X\), \(\dim_k T_pX\ge\dim_pX\). The point \(p\) is smooth if and only if \(\OO_{X,p}\) is a regular local ring, and regularity of the local ring is defined at every point of a scheme, closed or not.
The \(f_i\) in the Jacobian criterion generate the ideal \(I(X)\). Equations that cut out \(X\) only as a set do not suffice: \(V(x^2)=\{0\}\subseteq\AA^1\) has \(J(0)=(0)\), of rank \(0\ne1\), although \(0\) is a smooth point of \(\{0\}=V(x)\).
Over a perfect field, the Jacobian condition at \(p\) is equivalent to regularity of \(\OO_{X,p}\). Over an imperfect field, the Jacobian condition is smoothness, which is preserved by base change to \(\bar{k}\); regularity of \(\OO_{X,p}\) is not. Over \(k = \FF_p(t)\) the closed subscheme \(V(x^p - t) \subseteq \AA^1\) is the spectrum of a field, hence regular, but it is not smooth: after base change to \(k(t^{1/p})\) it becomes \(V((x - t^{1/p})^p)\), which is not reduced.
Dependencies
D-5LJUXDimension of a varietyuses
Backlinks
D-VARBLOWThe blowup of a variety at a pointrelated-toD-VARNCROSSNormal crossing singularitiesusesD-QJ5M9Normal domains and schemes, and their relation to regularityusesD-VARDUALThe dual variety of a projective varietyuses