The following theorem is from Hartshorne's book on ample subvarieties:
Some definitions:
The theorem:
This is Corollary 1.2:
This is prop 1.3: (The proof of this one is very long not sure if I can copy paste pages of proof here)
Hartshorne always assumes he is working on algebraically closed fields and sometimes the assumption is not necessary. I was wondering whether this result is true if we drop the assumption that we are working on an algebraically closed field or not? I wasn't able to find where it is used exactly but it could be implicit somewhere. If true without the algebraically closed-ness I wonder whether there is a reference for it?