Smooth variety contained in another smooth variety I apologize if this is a trivial question.  If $X$ is a smooth irreducible codimension two subvariety of projective space $\mathbb P^n$, then does there always exist a smooth irreducible codimension one subvariety $Y \subset \mathbb P^n$ with $X \subset Y$ ?
 A: The obvious thing to do is to choose $H$ a general hypersurface containing $X$.  In other words, choose a general global section of $I_X \otimes O_{P^n}(k)$ for $k \gg 0$.  
Certainly $H$ is smooth away from $X$ by Bertini.  I don't see why it should be smooth along $H$ though (unless of course, $X$ is a complete intersection).
In general, you are still ok locally, in other words a regular ring is always locally a complete intersection, so in a neighborhood of every point there is such a variety which is smooth near that point (they might not glue, or be smooth elsewhere though).  This follows from page 171 of Matsumura's Commutative Ring Theory.  See in particular 21.2(ii).
A: For $n = 3$ this is always true. It is also true when $X$ is a complete intersection.
Suppose $n ≥ 4$, and suppose that $Y$ exists; then by a well known result of Lefschetz $X$ is a hyperplane section of $Y$, and so $X$ is a complete intersection. Now, for $m = 4$ there are subvarieties of $\mathbb P^4$ that are not complete intersections (for example, the image of a generic projection $\mathbb P^2 \to \mathbb P^4$ of a quadratic Veronese embedding of $\mathbb P^2 \subseteq \mathbb P^5$), so the answer is negative for these. For $m ≥ 5$ the existence of codimension 2 subvarieties that are not complete intersections is a big open question; for $m ≥ 7$ it is a particular case of a conjecture of Hartshorne that these should not exist.
