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I am having some confusion on the definition of quasi varieties. From my book that I am reading, the definitions are:

  1. A projective variety is a closed irreducible subset $Y \subset \mathbb{P}^n$ for some $n$. A quasi-projective variety is a non-empty open subset of a projective variety
  2. An affine variety is a closed irreducible subset of $\mathbb{A}^n$ for some $n \geq 0$. A quasi-affine variety is an non-empty open part of an affine variety.

There are some thing that confuses me here. For instance, is it a correct understanding that a quasi-affine variety is firstly a subset of an affine variety and then the complement of some closed subset? If yes, is the complement considered in the affine $n$-space $\mathbb{A}^n$ or in affine variety? I suppose the same understanding holds for a projective variety?

Also, is there any difference in the wordings "open part" and "open subset"?

I would also appreciate if someone could provide some example, trivial or not.

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  • $\begingroup$ As far as I know, 'open part' is the same thing as an open subset. $\endgroup$
    – Daniel
    Oct 17, 2022 at 19:15

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Regarding your first question: The Zariski topology on any closed subvariety is induced from the ambient space, so for example a quasi-affine variety can be obtained by removing a closed subset either from the affine variety or from the affine space.

Open subset=open subscheme, in algebraic geometry, and open part is just a casual way to say the same thing. People usually treat open subschemes and open subsets equally since an open subset naturally has the unique scheme structure, by restricting the structure sheaf $\mathcal{O}_X$. However this is different in the case of closed subschemes: a closed subset could have many scheme structures, so when we need to talk about closed subschemes, we usually speak out the structure. For example, a closed subset with the reduced scheme structure is a common description of a closed subscheme.

The point here is: quasi-projectives (resp. quasi-affines) are usually not projective (affine) but good enough. Affines are all quasi-projective. But an affine variety is not projective unless in dimension $0$ case. An example of quasi-affine but not affine variety can be: $\mathbb{A}^2-\{p,q\}$.

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  • $\begingroup$ Thanks for your answer. So basically you are saying have an induced zariski topology on the actual variety, but I dont see how that implies the fact that a quasi affine variety can be obtained by removing a closed subset from either (its two different closed sets, can one exist while the other doesn't?) Could you please as elementary as possible example the "good enough" statement. I have seen it before but not quite sure I grasp it. Especially the part that affines are all quasi but not the other way around? Thanks $\endgroup$ Oct 19, 2022 at 12:10
  • $\begingroup$ Thinking about it, an affine variety is always quasi since we can consider it as a subset of itself and then removing the closed set given by the constant polynomial $1$. Is this correct? $\endgroup$ Oct 19, 2022 at 12:36
  • $\begingroup$ @BillDasque: Yes, empty set is Zariski closed (certainly this is also a necessary condition to ensure the Zariski topology is really a topology). What I mean by "good enough" here is that quasi-projectives and quasi-affines are for example separated (morally Hausdorff in topology). Also the conditions of projectivity and affineness are too strong in general applications: we often need to deal with open subvarieties of them, so why not give them a name. $\endgroup$ Oct 19, 2022 at 16:04
  • $\begingroup$ One thing I should mention is that in the textbook you are reading, they also identify closed subset=closed subvariety with the (only) reduced scheme structure. In general, schemes can be non-reduced. But textbooks at the beginning usually try to be friendly and not talking about the most general case. $\endgroup$ Oct 19, 2022 at 16:07

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