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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 $\mathbb{P}^n$ ...

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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
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