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Let $X$ be a smooth projective complex variety. Let $Y \subset X$ is a singular closed subvariety of $X$. Does there exist a birational morphism $\pi: \widetilde{X} \to X$, such that the proper preimage $\widetilde{Y}$ of $Y$ is smooth? In addition, can we obtain $\pi$ as a composition of blow-ups with smooth centers?

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    $\begingroup$ This is called "embedded resolution of singularities". A good reference is Kollar's book. $\endgroup$
    – Sasha
    Sep 11, 2016 at 17:04
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    $\begingroup$ I don't know who that author is whom Sasha mentions ... :), but there is a very good book on the subject written by Kollár(!): press.princeton.edu/titles/8449.html $\endgroup$ Sep 11, 2016 at 19:21
  • $\begingroup$ If there is an embedding of X into a regular variety Y. A strong desingularization of X a proper birational morphism from a regular variety $f:Y' \rightarrow Y$ satisfying the following conditions : 1) The transform $X′$ of $X$ is regular and transverse to the exceptional locus of the resolution morphism. 2)The restriction $f_{X'}:X' \rightarrow X$ is an isomorphism away from the singular points of X (thus they have the same function field). $\endgroup$ Sep 11, 2016 at 22:48
  • $\begingroup$ The construction of $Y′$ is given by a succession of a blowing up of regular closed subvarieties of $Y$ or more strongly regular subvarieties of $X$, transverse to the exceptional locus of the previous blowings up. $\endgroup$ Sep 11, 2016 at 22:48
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    $\begingroup$ Hironaka proved that the strong desingularization always exists for varieties of characteristic 0, but for varieties of characteristic p, this problem stills open in dimensions 4 or more and the construction of Hironaka was improved and actually we know that the construction of $Y′$ is functorial for smooth morphisms to $Y$ and embeddings of $Y$ into a larger variety and we know also that the morphism from $X′$ to $X$ does not depend on the embedding of $X$ in $Y$. $\endgroup$ Sep 11, 2016 at 22:51

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