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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.

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    $\begingroup$ Check out the general Jacobi's criterion in EGA or in Liu's book. $\endgroup$ – Martin Brandenburg Jan 23 '12 at 15:33
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The Jacobian Criterion can be applied to any kind of point on a projective (or affine) variety, closed or not. In the non-closed case one has to adapt the requirement for the rank of the Jacobian appropriately. The field $k$ should be perfect.

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