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Different authors seem to have different conventions when they define the term affine variety (similarly projective variety). For the purposes of this question let us stick with the affine case, and let us work over an algebraically closed field. For example:

  • In Harris's Algebraic Geometry: A First Course, an affine variety is the zero set in the affine space, of a collection of polynomials. So, it is just a closed subset of the affine space under the Zariski topology. He calls an irreducible closed subset, an irreducible affine variety. (A similar convention is used in the book by Cox, Little and O'Shea)
  • In Hartshorne's Algebraic Geometry, a closed subset of the affine space is called an affine algebraic set, and an irreducible closed subset is called an affine variety.
  • In the recent book Algebraic Geometry I: Schemes With Examples by Goertz and Wedhorn, the authors use the terms affine algebraic set, and irreducible affine algebraic set for closed and irreducible closed subsets of the affine space respectively. They reserve the term affine variety for a space with functions that is isomorphic to a space with functions associated to an irreducible affine algebraic set (so, this is more in the spirit of Hartshorne).

While it is usually clear from the context, what the authors of a particular book mean, when they use the terms above in bold, why are there different terminologies? Is there a consensus among mathematicians today, as to what they mean when they use the term affine algebraic variety?

Rankeya
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    Dear Rankeya, it is worse than that: it is not clear if affine varieties should be defined over algebraically closed fields or not – Georges Elencwajg Nov 12 '11 at 18:47
  • Thanks for pointing this out. After your and Justin Campbell's comments, I have edited the question so that we consider the case of an algebraically closed field. – Rankeya Nov 12 '11 at 19:07

1 Answers1

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First, the definitions you list give only affine varieties of dimension at most $1$ (i.e. finite sets and curves), along with the affine plane. To get a general definition replace "affine plane" with "affine space."

Second, there is no consensus on whether varieties are irreducible by definition: one simply has to be aware of the convention used by a particular author.

Third, one must be a bit careful about thinking of an affine variety as a closed subset of affine space with the Zariski topology: this is only correct if one remembers either the embedding or the polynomial functions on the variety. For example, all curves are homeomorphic as topological spaces since they are simply infinite sets with the cofinite topology, but one should distinguish between e.g. singular and nonsingular curves, so this is clearly not satisfactory.

Fourth, thinking of affine varieties as embedded in affine space is aesthetically displeasing (at least to people like me) because the coordinates are not "intrinsic" to the variety structure. My preferred definition would be a topological space equipped with spaces of functions for every open set (i.e. a sheaf of functions) which is isomorphic to the (maximal) spectrum of a "nice" algebra, or, a little less abstractly, isomorphic to a closed subset of affine space with the usual polynomial functions. Which perspective you take depends on your taste and what you wish to do with algebraic geometry.

Edit: (in response to Georges's comment) all of this discussion applies over algebraically closed fields. When one works with general fields things get more complicated, so it is best to understand the situation over algebraically closed fields first.

Justin Campbell
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  • I apologize for the first error. I meant affine space, but ended up writing affine plane for some reason. It has been edited. Thanks for your answer. – Rankeya Nov 12 '11 at 19:04
  • @Rankeya: I hope I'm not being rude, but unless you're waiting for more answers, the appropriate way to thank me for my answer is to accept it. Also, I would be happy to add more if you have further questions or if I haven't satisfactorily addressed your concerns in the question. – Justin Campbell Nov 13 '11 at 17:51
  • I was waiting for more answers. I often get delayed responses to the questions I have asked. – Rankeya Nov 13 '11 at 19:35
  • This does not mean you have no answered the question satisfactorily. I think your answer is lucid and to the point. – Rankeya Nov 13 '11 at 19:36
  • Thanks, I know people sometimes forget about questions after a day or two so I just wanted to make sure... – Justin Campbell Nov 13 '11 at 20:13