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This is a follow up question to my previous one, thanks to Karl and Angelo for their answers, and to the commenters: Smooth variety contained in another smooth variety

If $X$ is a local complete intersection (irreducible) subvariety of codimension two in $\mathbb P^n$ (we may assume $X$ is smooth if it helps, but I don't think it's necessary) then we know by Bertini's theorem there exists an irreducible hypersurface $Y$ containing $X$. Can we choose $Y$ such that $X$ is a Cartier divisor on $Y$ ?

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    $\begingroup$ It is necessary for $X$ to be smooth, see page 179 of Hartshorne. $\endgroup$
    – Christopher Perez
    Commented Feb 23, 2012 at 13:12
  • $\begingroup$ I had an answer below, but I misread the question (I didn't see the word local). It has been deleted. $\endgroup$
    – Karl Schwede
    Commented Feb 23, 2012 at 14:43

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The Grothendieck-Lefschetz theorem implies that if $Y$ is complete intersection in $\mathbb P^n$ of dimension at least 3, any Cartier divisor on $Y$ is obtained by intersecting with a hypersurface ($Y$ does not have to be smooth). So if $Y$ exists $X$ is a complete intersection, and the previous discussion applies.

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