Let $U$ be a smooth quasi-projective variety. Does there always exist a smooth compactification of $U$? If not always when can we have smooth compactification? In particular, suppose $X$ is a singular projective variety and $U$ is the smooth locus. The question is does there always exist a smooth projective variety $Y$ containing $U$? If we add the condition that $U$ is open in $Y$ is $Y$ uniquely determined upto isomorphism?
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The main idea is this: You can always find an $X$ such that $X\supseteq U$ and $X\setminus U\supseteq \Sigma := \mathrm{Sing} X$. Then in characteristic zero apply Hironaka's resolution theorem, which says that there exists a resolution of singularities $\pi:Y\to X$ such that $\pi$ is an isomorphism over $X\setminus \Sigma\supseteq U$. In particular, $\pi^{-1}: U\hookrightarrow Y$ gives an embedding. In positive characteristic resolution is not known in general and similarly this embedding result is not known either (although I am not saying that knowing this would prove the existence of resolutions). There is actually a newer, expanded version of Kollár's paper in book form: Lectures on Resolution of Singularities. And of course it is not unique as long as $U$ itself is not projective, since as you can always blow-up $Y$ outside of $U$. |
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In characteristic 0, yes, yes, no. Check out Kollar's paper: http://arxiv.org/abs/math/0508332 In characteristic p, unknown. |
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