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Sep 2, 2013 at 20:34 comment added ivainsencher I think that mod a typo you've clarified it all.
Sep 1, 2013 at 17:19 comment added Allen Knutson Sorry, you didn't use $C'$ in your question, and I didn't notice you were using it in your "OK if $C$ is integral" section. My $C'$ is not your $C'$. Since you assume $Z$ and $F$ disjoint, your question is equivalent to "Does $C$ have a subscheme only finitely much smaller?" to which one then adds points from $F$.
Aug 29, 2013 at 15:39 comment added ivainsencher $C'$ is assumed to be a subscheme of $Z$, not necessarily of $C$ proper. Of course if this were so, the equality of hilbpols would imply $C=C'$ at once. The difficulty is how to control possible embedded points of $C'$.
Aug 28, 2013 at 23:43 history answered Allen Knutson CC BY-SA 3.0