Canonical scheme structure on the singular locus of a variety

Let $X$ be subvariety of affine space $\mathbb{A}_{k}^n$, where $k$ is a field, and suppose $X$ is given by equations

$$X:(F_1=\cdots = F_m=0)\subset \mathbb{A}^n.$$

Then $X$ is singular at $p\in X$ if

$$\text{rank}(a_{ij}(p))<d,$$

where $d=n-\text{dim}(X)$ (i.e., the codimension of $X$ in $\mathbb{A}^n$), and $(a_{ij})$ is the matrix valued function

$$(a_{ij})=\frac{\partial F_i}{\partial x_j}.$$ Thus the ideal generated by the defining equations for $X$ along with the $d\times d$ minors of the matrix $(a_{ij})$ define a natural scheme structure on the singular locus of $X$ (as a subscheme of $\mathbb{A}^n$). Let's call this scheme $S_X$. My question then is the following:

Is $S_X$ intrinsic to $X$ (i.e., independent of the embedding $X\hookrightarrow \mathbb{A}^n)$?

• One high-tech way to see this is to note that $S_X$ is the same as the locus of points where $\Omega_X^1$ has rank less than $\dim X$. Since the sheaf $\Omega_X^1$ is an invariant of $X$, we're done. Apr 11, 2016 at 14:38
• But $S_X$ is in general a non-reduced scheme, not just a locus of points. Apr 12, 2016 at 1:29

The answer to the question is yes. There is a characterization of singular points of a variety in terms of the local ring of the point $x\in X$: this is the local ring obtained by localizing the coordinate ring of an (affine) neighborhood open of $x$ at the maximal ideal (or at the prime ideal in the case you prefer the language of schemes) corresponding to $x$. Let's denote $A$ this local ring and $\mathfrak{m}$ its maximal ideal.

Then the point $x$ is simple (non-singular on $X$) if and only if $A$ is a regular local ring, that is, if the dimension of the $A/\mathfrak{m}$-vector space $\mathfrak{m}/\mathfrak{m}^2$ equals the Krull dimension of $A$.

This discussion is detailed in the book Basic Algebraic Geometry (I have the old single volume version), by Shafarevich.

• How does this help you put a scheme structure on the singular locus?
– Hoot
Jul 2, 2016 at 3:35
• Pick an embedding $X\hookrightarrow \mathbb{A}^n$. The ideal defining $S_X$ is the ideal generated by the equations of $X$ plus the $d\times d$ minors of the jacobian matrix. This gives you the scheme structure for $S_X$. Once it is agreed that $S_X$ does not depend on the embedding we're done. Jul 2, 2016 at 16:58
• I think that last bit is what is at issue here.
– Hoot
Jul 2, 2016 at 18:58