How can I check if a curve with four variables (in $\mathbb{P}^3$) is non singular? For example, the curve over $\mathbb{F}_2$ given by
\begin{cases} y^2z+z^2y+xzt+yzt+xt^2+yt^2= 0 \\ x^2+yz+xt+yt+zt+t^2=0. \end{cases}
In two variables (affine case) and in three variables, I can use the Jacobi Criterion, but I don't know how I can verify in my case. I think I have to look at the Jacobian matrix $$J=\left( \frac{\partial f_i}{\partial x_j} \right)_{1 \le i \le 2 \\ 1 \le j \le 4},$$
(where $f_1 = y^2z+z^2y+xzt+yzt+xt^2+yt^2$, $f_2 = x^2+yz+xt+yt+zt+t^2=0$ and $x_1=x$, $x_2=y$, $x_3=z$, $x_4=t$) and verify that J applied in $P$ has rank $n-1$.
Thanks in advance.