Dear Pete, thanks a lot for the reference and those keywords. (In your notes, the relevant result Corollary 111 has a broken reference to an earlier result (Proposition ??).)

Dear Pete, do you know a convenient reference for the statement in the first sentence of your second paragraph? A bit of Googling reveals it is called Steinitz' (third) theorem; I found Steinitz' original paper, but I would like something a bit more digestible to refer to.

@Mohan: as I mentioned in the question, I am not asking about isomorphism as varieties over $\mathbf C$. Donu's example shows what can happen when we allow this wider class of morphism.

@DonuArapura: right, that's the kind of situation I had in mind to convince myself this isn't completely crazy!