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](https://mathoverflow.net/questions/89229/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 $Y$ ?



Codim 2 subvariety Cartier on a hypersurface containing it
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.


