I am having some confusion on the definition of quasi varieties. From my book that I am reading, the definitions are:
- 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
- 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.
Best Answer
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: $\mathbb{A}^2-\{p,q\}$.