I need to prove that a point $P$ in a projective curve $C$ is non-singular if and only if its local ring $\mathcal{O}_P$ in a Discrete Valuation Ring. I know this is true on a affine variety. Can you help me ?
2025-01-13 11:54:32.1736769272
Affine variety versus projective
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Remove a different point Q from the projective curve C. Them you get a a new affine variety (curve). Now the local ring at P both for this affine curve and the given projective curve are the same (or isomorphic).
That is local ring of a point of a variety depends only on (any) affine open subset containing that point. In the case of manifolds the local ring is defined by using germs of differentiable/analytic functions which depends only on the open subset containing that point.