# Checking smoothness of a projective variety

Let $$X$$ be the proj variety defined by $$X= Z(x_1^2-x_0x_2, x_3^2-x_2x_4)$$ I want to check the singular points of that. The Jacobian is given by: $$\begin{bmatrix} -x_2 & 2x_1 & -x_0 & 0 & 0\\ 0 & 0 & -x_4 & 2x_3& -x_2 \end{bmatrix}$$ Now if $$X_2$$ is not zero then the minor with: $$det\begin{bmatrix} -x_2 & 0\\ 0 & -x_2 \end{bmatrix}$$ is not zero so we have maximal rank and so these points are smooth. Now if $$x_2=0$$ then $$x_1=x_3=0$$ and so the jacobian becomes: $$\begin{bmatrix} 0 & 0& -x_0 & 0 & 0\\ 0 & 0 & -x_4 & 0& 0 \end{bmatrix}$$ which has rank 1. So in this case we have singular points. Now I want to find the tangent cone at each singular points. Now I have some troubles but i think that since $$TC = Z(I^{in})$$ is given by the initial ideal, but $$I$$ is homogeneus so i think that $$TC=X$$ it is correct? Thanks for suggestions!