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Let $k$ be an algebraically closed field, and $V$ a normal (irreducible) affine variety over $k$. Does there necessarily exist a closed immersion $V \hookrightarrow \mathbb{A}^n$ of $V$ into affine space such that the closure of $V$ in projective space $\mathbb{P}^n$ is normal?

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up vote 6 down vote accepted


Take a closure $\bar X$ of $X$ in some projective space. We can write $\bar X= X\cup D$, with $D$ ample. Normalize $\bar X$ to get new projective variety $\pi:\tilde X\to \bar X$. The preimage $\pi^{-1}D$ is ample with complement $X$ because $\pi$ is finite. So $\tilde X$ can be re-embedded in another projective space $\mathbb{P}^N$ so that $\pi^{-1}D$ is set theoretically the intersection of $\tilde X$ with a hyperplane $H$. Under the embedding of $X\subset \mathbb{A}^N=\mathbb{P}^N-H$, the closure is $\tilde X$ which is normal.

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Nice! I assume that $D$ is taken as a Cartier divisor in this setting, since it is clearly locally principle but $\bar{X}$ may not be nonsingular in codimension 1. – Charles Staats Sep 18 '10 at 15:27
Yes, right $D$ is a Cartier divisor. – Donu Arapura Sep 18 '10 at 15:35
An affirmative answer here has finally allowed me to get some sort of answer to – Charles Staats Sep 18 '10 at 19:14

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