When can we apply the Jacobi's criterion for the projective variety $V(f_{1}, \ldots, f_{r}) \subset \mathbb{P}^{n}$ in order to find the singularities of the scheme $\mathrm{Proj} \left( k[x_{1}, \ldots, x_{n+1}] / (f_{1}, \ldots, f_{r}) \right)$?
In Hartshorne's book Algebraic Geometry, Proposition II.2.6, we have a fully faithful functor from the category of varieties over $k$ to the category of schemes over $k$, but it seems to provide information only for the closed points of the scheme.
Thank you.