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What are the differences between the terms prevariety, variety, abstract variety, and algebraic variety? Sometimes I see these terms used in the literature without explicit definitions or references, and it is not clear that all of them have standardized definitions.

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The last three are the same. The first is more general in that it may not be separated. If you tell us more about your background, someone may give a more expansive answer. –  Donu Arapura Apr 20 '11 at 15:02
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I don't think that this terminology is very standardized... but: I think that algebraic variety should certainly encompass the preceding two terms. The difference between variety and abstract variety is probably whether the thing is explicitly stuffed into affine/projective space or just regarded as a separated scheme of finite-type over a field. The "pre" prefix likely relaxes the separated condition. –  Ramsey Apr 20 '11 at 16:46
    
I agree with Ramsey, if someone says to me abstract variety, then I take it to mean he is definitely thinking about not-necessarily quasi-projective varieties. In particular, it means they are emphasizing the fact that it might not be quasi-projective. Of course, sometimes people say variety and only mean quasi-projective variety. –  Karl Schwede Apr 20 '11 at 18:41
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@bird1962: Sorry for putting words in your mouth. The original form of the question was tough on my eyes. If you want to revise it further, please follow the link at the top of this page to the "how to ask" page. –  S. Carnahan Apr 21 '11 at 5:59
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1 Answer 1

There is no standardized definition. Using one of these terms always implies there is a ground field $k$, and (pre)varieties are $k$-schemes of finite type, possibly with certain restrictions (separated, reduced, irreducible...).
As far as I know, "prevariety" and "abstract variety" are no longer in use. "Prevariety" used to mean "possibly nonseparated" (at least in Mumford's Red Book, where prevarieties are in addition irreducible and reduced). "Abstract" stands for "not embedded in projective space".
Regrettably, some authors use the term "(algebraic) variety" without bothering to define it. In general, such carelessness is an indication that the objects in question are assumed separated and reduced (other cases not being worthy of consideration), or even that $k=\mathbb{C}$ (for the same reason).

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