Talk:Algebraic variety

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Added links to the notions of stacks and algebraic spaces (further generalizing varieties), which are of course, not yet in existence.

Actually we do have pages on those topics, and I have modified the links to go there. Charles Matthews 08:39, 6 September 2005 (UTC)

[edit] Irreducibility

In many text books I have access to, an affine variety is simply the set of common zeros of a set of polynomials, NOT NECESSARILY IRREDUCIBLE. Pura 04:07, 1 January 2006 (UTC)

Books are not consistent on this. Other books require irreducibility. (I think Hartshorne requires irreducibility, for example.) -- Walt Pohl 07:14, 1 January 2006 (UTC)

See algebraic set. You need irreducibility for the definition of function field and hence of dimension of an algebraic variety. Charles Matthews 08:50, 1 January 2006 (UTC)

The text books that I happend to have are quite dated, so I'm not sure if which convension is more popular now. But the fact that this term is used inconsistently over different books is perhaps worth noting in the article, to avoid confusion. ---- Pura 19:40, 2 January 2006 (UTC)

I agree. Throughout the texts that I learned about algebraic varieties, any closed set in the zariski topology was referred to as a variety. I think Hartshorne restricts the term to irreducible sets, but there should at least be some mention that the term is not completely agreed upon. See Shafarevich for example (I believe). CraigDesjardins

I have taken this to Wikipedia:WikiProject Mathematics/Conventions: discuss on the talk page there. Charles Matthews 09:40, 23 January 2006 (UTC)

[edit] Algebraic manifolds

I think one has to be careful with this. Certainly a non-singular point of a real or complex algebraic variety implies a local chart, so smooth varieties are manifolds. IIRC, the comverse need not be true, though.

Further, I think the definition may start from the wrong end. really. For example, for a real manifold that is embedded, algebraic manifold means that there are polynomial equations defining it.

Charles Matthews 16:06, 24 March 2006 (UTC)

Yep, a bit sketchy at the moment, still trying to work on it. I've now moved things to Algebraic manifold where it can be worked on a bit. I've only got a few contridictory web references at the moment. Oleg requested the article so I had a stab. --Salix alba (talk) 17:32, 24 March 2006 (UTC)

[edit] Whitney's theorem

Theres a theorem by Witney (Brooker and Larden 1975):

Any closed set in Rⁿ occurs as the solution set of f^-1(0) for some smooth function f:Rⁿ→R.

Hence any manifold embedded in Rⁿ can be defined as the solution set of a smooth function. This does not automatically imply that its a variety.

So my question is to what extent can manifolds be defined as varieties. I guess it would be posible to say some along the lines of almost all. Sard's lemma and Morse theory may also come into play. --Salix alba (talk) 09:55, 25 March 2006 (UTC)