Let $P\in Y=Z(f_{1},\cdots ,f_{s})$ be a projective variety. Then $Y$ is non singular at $P$ if and only if rank of the matrix $\Vert\partial f_{i}/x_{j}(P)\Vert=n-\dim{Y}$.
I know of the statement for affine varieties, and I am trying to prove it for projective varieties. This is what I have so far.
If $P\in \mathbb{P}^{n}$ then $P$ is in some open set $U_{i}=\{(a_{0},\cdots ,1,\cdots a_{n})\in\mathbb{P}^{n}\}$ where the $1$ appears in the $i^{th}$ spot. WLOG we may assume $i=0$ is this case. So $X\cap U_{0}$ is an affine variety. Defined by $$X\cap U_{0}=Z(f_{1}(1,x_{1},\cdots x_{n}),\cdots ,f_{s}(1,x_{1},\cdots x_{n}))$$ Since this is affine we have that $X\cap U_{0}$ is nonsingular iff and only if the jacobian criterion is satisfied. I don't really see where to proceed from here since I don't know how the partials of the dehomoginzed polynomials compares to that of the homogeneous polynomials. Any suggestions would be very appreciated.