I first asked this question on Math StackExchange but no answers were given.

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 $X$?