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.

  • 1
    $\begingroup$ Check out the general Jacobi's criterion in EGA or in Liu's book. $\endgroup$ – Martin Brandenburg Jan 23 '12 at 15:33

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.