Branch locus of a projection of algebraic set

Question

Let $X$ a algebraic cone in $\mathbb{C}^n$ with $\dim_0 X=p$.

def.: if $f:A\to B$ is smooth map between smooth manifolds, then $br(f)$ is the points $x$ that $df_x$ is not surjective.

def.: if $\pi: \mathbb{C}^n \to \mathbb{C}^p$ is a linear projection, then $br(\pi|_X)=Sing(X)\cup br(\pi|_{Reg(X)})$.

Are there coordinates $(z_1,...,z_n)$ in $\mathbb{C}^n$ (and a linear projection $\pi: \mathbb{C}^n \to \mathbb{C}^p$) such that $br(\pi|_X)=Sing(X)$ (i.e., $br(\pi|_{Reg(X)})=\emptyset$) and $\pi^{-1}(0)\cap X=\{0\}$?

Answer 1

If $\dim C_4(X)=\dim X$, then this is hold, look Chirka, Complex analytic sets, section 9.4 proposition 1.