I have a question in algebraic geometry that I would like to ask:
Let $ \mathbb{k} $ be an algebraically closed field. Is there a property $ P $, phrased in the language of schemes, such that every $ \operatorname{Spec}(\mathbb{k}) $-scheme $ (X,\mathcal{O}) $ is isomorphic to the $ \operatorname{Spec}(\mathbb{k}) $-scheme associated with a quasi-projective variety over $ \mathbb{k} $ if and only if $ X $ satisfies $ P $?
By isomorphism, I mean one between locally ringed spaces.
By quasi-projective variety, I mean a Zariski-open subset of a projective variety (i.e. an irreducible projective algebraic set) in $ {\mathbf{P}^{n}}(\mathbb{k}) $ (the projective $ n $-space over $ \mathbb{k} $) for some $ n \in \mathbb{N} $.
I am guessing that $ P $ would have to include properties like ‘integral’, ‘separated’ and ‘of finite-type’, but I am not sure if these are sufficient.
Thank you for your help!