Let $J\subset \mathbb{C}[x_1,\dots,x_n]$ be an ideal such that $J\subset (x_1,\dots, x_n)$ we can define the Zarisky tangent space of the Scheme $X = Spm(\mathbb{C}[x_1,\dots,x_n]/ J)$ at $0$ by $T^{\text{Zar}}_0 = \mathbb{V}(J_1)$, where $J_1 = (\lbrace f_1, f\in J\rbrace)$ and $f_1$ is the linear part of $f$. There is another definition using formal path
$$T^{\text{Zar}}_0 = \lbrace \gamma'(0), \ \gamma(t)\in \mathbb{C}[[x]]^n,\ \gamma(0) = 0,\ f\circ \gamma(t) = 0 \mod t^2, \ \forall f\in J \rbrace.$$
I try to make the same thing at the second order so for $f\in J$ decomposing in homogeneous component $f = f_1 + f_2 + \cdots$, I set $f_{1,2} = f_1$ if $f_1\neq 0$ and $f_{1,2} = f_2$ otherwise. I define $J_2 = ( \lbrace f_{1,2}, f\in J \rbrace)$ and $QC_0 = \mathbb{V}(J_2)$. This definition is inspired from the definition of the tangent cone and seems good. Now I try do do the same with formal paths
$$ \mathcal{QC}_0 = \lbrace \gamma'(0), \ \gamma(t)\in \mathbb{C}[[x]]^n,\ \gamma(0) = 0,\ f\circ \gamma(t) = 0 \mod t^3, \ \forall f\in J \rbrace.$$
We have $\mathcal{QC}_0 \subset QC_0$ but I am not able to proove the other inclusion and even not sure it is the same. Because if $\gamma(t) = t\gamma_1(0) + t^2\gamma_2(0) + \cdots$, we have $f\circ \gamma(t) = tf_1(\gamma_1(0) + t^2(f_1(\gamma_2(0)) + f_2(\gamma_1(0))) \mod t^3$. Hope someone can help me with this question.
An other question is if the ideal $J_2$ is radical and $J_2 \neq (0)$, is it still possible that three components pass through the point $(0,\dots,0)$. I hope it's not but I am infortunaly not able to proove it.