MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Differentiable manifolds can be described in terms of local charts to open subsets of $\mathbb{R}^n$ and transition functions that are diffeomorphisms. Trying to put $\mathbb{A}^n$ (over an algebraically closed field, say) in place of $\mathbb{R}^n$ and isomorphisms of open subvarieties of $\mathbb{A}^n$ in place of local charts of course will not reproduce all the algebraic varieties, even the nonsingular ones, as they have a rich local structure (the stalk of the structure sheaf at a point fully determines its birational equivalence class etc.).

My question is

What about the varieties that can be described in this way? At least for the complete ones, is there a characterization or a classification?

Projective spaces and grassmannians belong to this class. Clearly they must be smooth, and birational to $\mathbb{P}^n$ ...

share|cite|improve this question
If I understand correctly, you are asking which smooth, proper varieties have the property that every point has a neighborhood isomorphic to an open in affine space, or equivalently, the local ring of every point is a localization of a polynomial ring. Obviously every such variety is rational. Conversely, every rational variety of dimension 1 or 2 has this property. But the last that I heard, this property is open already for rational threefolds, even for the blowing up of a curve in projective 3-space. – Jason Starr Jul 10 '11 at 22:59
Isn't this already a good answer? Often I ask myself why comment-boxes are used for answers. – Martin Brandenburg Jul 11 '11 at 7:34
@Jason: I agree with Martin that you could post your comment as an answer. – Qfwfq Jul 11 '11 at 11:44

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.