(Jacobian criterion)
Let \(k = \kbar\) and let \(X = V(f_1, \dots, f_r) \subseteq \AA^n_k\) be of pure dimension \(n - r\). Then \(X\) is smooth at a closed point \(p\) exactly when the jacobian matrix \[ J(p) = \qty{ \partial f_i / \partial x_j (p) }_{i,j} \] has rank \(r\). In general \(X\) is smooth at \(p\) of dimension \(n - \rank J(p)\) when the two agree, and \(\rank J(p) \leq r\) always, with strict inequality exactly at the singular points.
The criterion is the standard computational tool in the topic: write the equations, differentiate, and solve \(J = 0\) together with the equations themselves.
A point of \(V(J)\) is singular only if it also lies on \(X\).
The hypothesis \(k = \kbar\) is load-bearing and its failure is the same characteristic-\(p\) inseparability seen elsewhere. So is the pure-dimension hypothesis: for \(X = V(xy, xz) \subseteq \AA^3\), a plane meeting a line, the jacobian rank is wrong along the whole line because the codimension is not constant.
The intrinsic statement behind the criterion is that \(\mfm_p/\mfm_p^2\) has dimension \(\dim \OO_{X,p}\), that is, \(\OO_{X,p}\) is a regular local ring; the jacobian just computes the dimension of that cotangent space from a presentation. That equivalence between smooth over \(\kbar\) and regular is what the criterion is really testing.
Dependencies
D-MORSMSmooth morphisms and relative dimensionuses