There's a well-known correspondence (traditionally called Hartshorne-Serre) between codimension 2 smooth subvarieties $S\subset X$ of a smooth algebraic variety $X$ and certain rank two vector bundles on $X$. Is there anything similar for higher codimensional subvarieties of $X$, i.e. codimension 3 and more?
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9$\begingroup$ The title made me think of Hartshorne and Serre sending letters to one another on 6-codimensional paper! $\endgroup$– MTSCommented Jul 16, 2013 at 1:25
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3$\begingroup$ To my knowledge, the problem of writing letters in codimension higher than one is still open... $\endgroup$– IMeasyCommented Jul 16, 2013 at 8:08
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2$\begingroup$ @MTS: conversely, one day I saw a fat book in the library entitled "Grothendieck--Serre correspondence". I didn't look inside, but instead spent quite a while trying to guess what the objects involved in this correspondence could be... $\endgroup$– user5117Commented Jul 16, 2013 at 14:05
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1$\begingroup$ @Artie: The objects involved were missives :) $\endgroup$– Jason StarrCommented Jul 16, 2013 at 14:46
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3 Answers
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To get some idea of the difficulty in generalizing Serre's correspondence to higher codimension, bear in mind that what makes a resolution $0 \rightarrow \mathcal{F} \rightarrow \mathcal{E} \rightarrow \mathcal{I}_{Z|X} \rightarrow 0$ plausible when $\mathcal{F},\mathcal{E}$ are vector bundles on $X$ and $Z \subset X$ is smooth of codimension 2 is that $Z$ is locally Cohen-Macaulay. If $Z$ is assumed to be locally Cohen-Macaulay of possibly higher codimension, then $\mathcal{F}$ is at best reflexive, and even if we pass to a resolution of $\mathcal{I}_{Z|X}$ by vector bundles, recovering $Z$ from a series of vector bundle maps is a highly nontrivial matter.
EDIT: As indicated below by Libli's informative answer, what is more relevant is that $Z$ is at least locally Gorenstein (which in codimension 2 is equivalent to being a local complete intersection).
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In codimension 3, you can hope for your varieties to be cut out (scheme theoretically) by the Pfaffians of an alternating map of vector bundles on $X$.
If you assume that $S$ is subcanonical, which you have to assume for Serre's correspondence (by the way, traditionnally, this is not called Hartshorne-Serre, but only Serre's correspondence) that $X = \mathbb{P}^N$ and some small extra-assumptions on $S$, then it is a theorem of Walter that $S$ is cut out by such Pfaffians (see : Walter, Pfaffians subschemes, Journal of Algebraic Geometry, 1996).
If $X$ is not the projective space, you can still hope for some structure theorems, but they are more difficult to explain (see : http://arxiv.org/pdf/math/9906170.pdf).
In codimension $4$ there is this : http://arxiv.org/pdf/1304.5248.pdf (which seems to me even much more complicated) and after (codimension $5$ or bigger) I guess nothing is known.
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As far as I have ever heard (and I have asked around), there is no general analogue of Serre's construction in higher codimension.