Lagmental Vicfred

The Jacobian Criterion Detects Smooth Complete Intersections by Vicfred

Last updated: Mon 10 August 2026

tags: mathematics, smoothness, singularities, etale morphisms

For a codimension-r complete intersection, rank r of the Jacobian characterizes smooth points under standard hypotheses. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

The data

For \(X=V(f_1,\ldots,f_r)\subseteq\mathbf A^n\), the Jacobian matrix \(J_p=(\partial f_i/\partial x_j)(p)\) controls tangent dimensions. Smoothness asks for the expected rank after passing to the residue field.

$$ X=V(f_1,\ldots,f_r)\subseteq\mathbf A_k^n $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ p\ \text{smooth}\Longleftrightarrow\operatorname{rank}J_p=r $$

Derivation

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ \dim_kT_pX=n-\operatorname{rank}J_p,\qquad\begin{cases}=n-r,&p\text{ smooth},\\>n-r,&p\text{ singular}.\end{cases} $$

Invariant content

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad X=V(f_1,\ldots,f_r)\subseteq\mathbf A_k^n,\\[5pt] \mathsf{C}\;&:\quad p\ \text{smooth}\Longleftrightarrow\operatorname{rank}J_p=r. \end{aligned} $$

Scope

A visually sharp point need not capture scheme-theoretic singularity, and characteristic can make every partial derivative vanish unexpectedly.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] p\ \text{smooth}\Longleftrightarrow\operatorname{rank}J_p=r \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Fri 08 May 2026. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.