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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!

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