I am part of a reading group that has completed a first course in algebraic geometry that culminated in a brief discussion of schemes and is now working through Hartshorne. This question is about his definition of isomorphism of varieties (in chapter I), and only deals with the classical case (i.e. I am not asking about varieties as schemes here, though that doesn't preclude answerers using the language of schemes if that clarifies things).
There is an isomorphism between the hyperbola and the punctured line ($\mathbb{A}^1 \setminus \{0\}$), and yet the hyperbola is affine while the punctured line is not (indeed, a quasi-affine variety is dense and open, and an affine variety is closed, so if a variety is quasi-affine and affine then it is the entire space). This seems odd to me because we would expect isomorphisms in the category of varieties to be defined in such a way that we preserve properties like "being affine" which seem to be quite important (indeed, many theorems are stated just for "affine varities" and not for quasi-affine varieties - e.g. isomorphism of varieties is equivalent to isomorphism of coordinate rings for affine varieties - but they can be extended to quasi-affine varieties if the latter are isomorphic to affine varieties).
Based on this, I would naively define "affine variety" to be "a variety isomorphic to a closed subset", rather than just a closed subset (this is the point we had a disagreement on). This has the advantage of making the category of affine subsets a much nicer subcategory of all varieties (since it is now closed under isomorphism).
The following question on MSE is relevant: There are quasi-affine varieties which are not affine - indeed, the accepted answer suggests that the "generally accepted definition" is to define affine varieties in the following way:
- Call the closed subsets of $\mathbb{A}^n$ affine subsets; and
- Declare the set of affine subsets to be closed under isomorphism (i.e. we throw in all the varieties isomorphic to the closed subsets of $\mathbb{A}^n$).
(which agrees with my naive definition). According to the OP (commenting on the answer), this is not mentioned in Hartshorne (and I certainly can't see it either).
My questions are, therefore,
- Is this definition (i.e. the "naive" closed-under-isom definition) the "generally accepted" definition of affine subset for working classical algebraic geometers - I am aware there might not be such a thing anymore! - or is the answer of the MSE question linked making a stronger statement than perhaps is correct?
- If the answer to (1) is "yes", is this mentioned in Hartshorne anywhere or is he being non-standard in his definition?
- Supposing we work with this definition, do we lose any "nice things" now that not all affine varieties are topologically closed? (This third question might be too unrelated so I am happy to open up a new question for it or delete it if it makes this question too broad.)
Prior research and other sources
- Shafarevich defines an affine variety to be "a quasiprojective variety isomorphic to a closed subset of $\mathbb{A}^n$" - this is p.48 of vol. 1
- Mumford (the Red Book) defines an affine variety to be a variety isomorphic to an irreducible algebraic subset (rather, he defines it to be a top. space $X$ together with a sheaf that makes $X$ isomorphic to an irreducible algebraic set with the natural sheaf) - this is defn I.6 on p.22
- On the other hand, Harris (his introductory book) defines an affine variety to be a zero set of a collection of polynomials (i.e. the same as Hartshorne but without the irreducibility condition), but only defines isomorpism between affine varieties or between projective varieties (not between varieties of all types) which seems non-standard (and does not mention the word quasi-affine at all).
- This question is also relevant, but does not directly address my question: I understand that there is disagreement in the textbooks about the definition of affine variety - I am more interested in knowing whether this disagreement is primarily notational or is a little deeper.
- It has been suggested to me (both by someone I asked in person, as well as in the accepted answer to this question, that the point of confusion here is that when we deal with classical varieties we are carrying around a lot of implicit baggage as we actually have to give an embedding into the ambient space. I know that this goes away when we pass to the theory of schemes, but I am not sure why it is a problem in the classical case since we can pass to a "coordinate-independent" form (by passing from the category of varieties-isomorphic-to-affine-varieties to the category of affine algebras, i.e. the coordinate rings). This just gives the same definition as the naive one above, and so I am still not sure why we "bother" carrying around the embedding.
If this question is too broad then please let me know, and I will try to tighten it up. Pointers to other relevant questions here that I've missed would also be appreciated!