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Let $X$ be an algebraic variety over an algebraically closed field $k$. Here a variety is an integral separated scheme of finite type over $k$ as in Hartshorne's book. Let $x \in X$ be a closed point. Assume that $x$ is a nonsingular point. Then $\mathscr{O}_{X,x}$ is a regular local ring. Then there exists an affine open neighborhood of $x$ such that every point $y \in U$ is a nonsingular point. I believe that $\Gamma(U, \mathscr{O}_X)$ is isomorphic to a localization of a finitely generated $k$-algebra.

Question: Is $\Gamma(U, \mathscr{O}_X)$ isomorphic to a localization of a polynomial ring over $k$?
If an answer is no, then can we choose $U$ such that $\Gamma(U, \mathscr{O}_X)$ isomorphic to a localization of a polynomial ring over $k$?

My feeling is that the answer is no to this question, but I do not have a counter example.

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Your $\Gamma(U,\mathcal{O}_X)$ is (by definition of finite type) a finitely generated $k$-algebra, not just a localization of one. –  Gregor Bruns Jun 27 '13 at 17:56
    
I see. If $R = \mathbb{C}[x]_{(x)}$ and $X= \hbox{Spec} (R)$. Then $X$ is not of finite type over $\mathbb{C}$? –  Youngsu Jun 27 '13 at 18:38
    
No, it is not. In fact, it is a rich source of counterexamples regarding schemes that are not varieties. –  Gregor Bruns Jun 28 '13 at 9:06
    
Thank you very much. –  Youngsu Jun 28 '13 at 13:59

1 Answer 1

If $X$ is irreducible this is true iff $X$ is birational to a projective space iff the function field of $X$ is $k(x_1, ... x_n)$ (since $X$ and any open $U$ share the same function field). Easy counterexamples include any smooth projective curve of positive genus (since genus is a birational invariant).

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I see. Thank you for the statement. Examples make sense to me. In the question since $X$ is a variety, it is irreducible. When you say $X$ is irreducible, is there any additional assumption on $X$? I wonder if you can give me a reference to the statement. Thank you. –  Youngsu Jun 27 '13 at 18:47

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