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 $X \subset Y$ ?
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$\begingroup$ Perhaps something like this would work? The ideal sheaf $I_Y(m)$ is generated by global sections for $m\gg 0$, by Serre's theorem. Fix such an $m>0$. Note that the general element in the linear system $H^0(I_Y(m))$ is smooth away from $Y$ by Bertini's theorem. So you are reduced to showing that the general element is smooth on $Y$. $\endgroup$– J.C. OttemCommented Feb 22, 2012 at 22:35
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$\begingroup$ For curves in $\mathbb{P}^3$, the following argument works: taking cohomology of the conormal sequence shows that global sections of $I_Y(m)$ give global generating sections of $N_Y^*(m)$, which is a vector bundle on $Y$ since $Y$ is assumed smooth. But general sections of $N_Y^*(m)$ do not vanish on on $Y$ (since such such loci have expected codimension 2). $\endgroup$– J.C. OttemCommented Feb 22, 2012 at 22:36
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1$\begingroup$ In case the complete intersection argument isn't obvious, if the variety is a complete intersection, then if one of the generators had vanishing derivatives somewhere on $Y$, the tangent space would have a dimension too high and form a singularity. $\endgroup$– Will SawinCommented Feb 22, 2012 at 22:44
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2 Answers
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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.
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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).
answered Feb 22, 2012 at 22:41
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$\begingroup$ Thank you Karl. So what you're saying is that I can find an irreducible hypersurface containing $X$ that is smooth away from $X$, even if $X$ is not a complete intersection? $\endgroup$– ParsaCommented Feb 23, 2012 at 7:27
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$\begingroup$ Parsa, that's true. Or for each point $z$ of $X$, you can find one that is smooth near $z$ (but then they don't glue). $\endgroup$ Commented Feb 23, 2012 at 12:34
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$\begingroup$ Perhaps one should also point out that you can always guarantee that $H$ is irreducible (also by Bertini). You can also guarantee that $H$ is normal I think, so at least $X$ is a Weil divisor on $H$. $\endgroup$ Commented Feb 23, 2012 at 12:45