4
$\begingroup$

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 $Y$ ?

$\endgroup$
  • 1
    $\begingroup$ It is necessary for $X$ to be smooth, see page 179 of Hartshorne. $\endgroup$ – Christopher Perez Feb 23 '12 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 Feb 23 '12 at 14:43
4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.