Jacobian criterion for projective varieties

Use Eulers Lemma for a homogeneous polynomial $f$, of degree $d$ $$\sum_{i=0}^{n}x_{i}\dfrac{\partial f}{\partial x_{i}}=df $$

Consider the affine cone $C(Y)$ in $\mathbb{A}^{n+1}$; it is defined by the ideal$(f_1,\dots, f_r)$ of $k[x_0,\dots,k_{n+1}]$. Choose a point $Q$ of $C(Y)$ which is equivalent to the point of $Y$ in question. The affine Jacobi criterion applied to $C(Y)$ shows that $C(Y)$ is smooth at $Q$ if and only if the rank of the $r\times (n+1)$-Jacobi matrix $\Big(\frac{\partial f_i}{\partial x_i}(Q)\Big)$, which may be replaced with the rank of $\Big(\frac{\partial f_i}{\partial x_i}(P)\Big)$, is $(n+1)-\dim C(Y)$. One can prove (i) $\dim C(Y)=\dim Y +1$, and (ii)$\dim T_Q(C(Y))=\dim T_P(Y)+1$; in particular, $Y$ is smooth at $P$ if and only if $C(Y)$ is smooth at $Q$. This implies the desired, projective Jacobi criterion. To prove (ii), one may assume $Q=(1,1,\dots,1)$ though an automorphism, and should show, using Linear Algebra, that the rank of the Jacobi matrix above does not decrese by removing some (or eventually, any) column from the matrix.