I am sorry if this question has been already asked, couldn't find anything similar myself. I have recently recalled this long standing open problem of whether every irreducible curve in $\mathbb{P}^3(\mathbb{C})$ is a set-theoretic intersection of two surfaces in $\mathbb{P}^3(\mathbb{C})$. I know that there are results in characteristic $p > 0$. My question is what is the best result regarding this problem known today? Some references would be appreciated.
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1$\begingroup$ Maybe I misunderstood the question, wouldn't twisted cubic be a counterexample? $\endgroup$– MichaelCommented Dec 17, 2014 at 22:12
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3$\begingroup$ Very little. I assume you meant connected, since a set-theoretic complete intersection curve is connected. @Michael, the twisted cubic is defined set-theoretically as the intersection of a quadric and a cubic. In $\mathbb{C}^3$, every smooth (actually local complete intersection) curve is a set-theoretic complete intersection, but in general, the problem is open. $\endgroup$– MohanCommented Dec 18, 2014 at 4:48
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$\begingroup$ @Mohan thanks, I've added irreducible to the statement of the question. I wonder just whether there are some general results. I've seen people constructing monomial curves that are set-theoretic intersection of two hypersurfaces, but I'd like something more general. $\endgroup$– shamovicCommented Dec 26, 2014 at 7:58
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$\begingroup$ @Michael the twisted cubic is a set-theoretical complete intersection. The singular quadric and a cubic cuts out a multiplicity two structure on it. $\endgroup$– myzhang24Commented Jan 21, 2020 at 0:58
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$\begingroup$ Could you please share what are the results in prime characteristic you are aware of? I googled, but couldn't find any? $\endgroup$– AuroraCommented yesterday
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1 Answer
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This question is wildly open still in characteristic zero.
We know some examples of curves that are s.c.i.’s, for example Rao’s classification of self-linked ACM curves, Hartshorne and Polini’s examples of curves on ruled cubic surfaces.
In Hartshorne and Polini’s recent paper on codepth, they reformulated the conditions for a curve to be an s.c.i. in terms of the new notion of codepth, and gave several necessary criteria.
One interesting necessary condition for $C$ in $P^3$ to be an s.c.i. is that $C$ must be contained in a hypersurface $X$ such that $X-C$ is affine. Thus an example of a smooth curve $C$ in $P^3$ not contained in any such hypersurface would give a counter example to the s.c.i. conjecture.
It is believed that the rational quartic is not likely an s.c.i, however no complete proof has been published yet. The best so far are results on the lower bounds of the degrees for the two equations should they exist.
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1$\begingroup$ $\mathbb{P}^3-X$ is affine for any hypersurface $X$. $\endgroup$– MohanCommented Jan 21, 2020 at 1:49
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$\begingroup$ @Mohan Indeed! It should be $X-C$, corrected. $\endgroup$ Commented Jan 21, 2020 at 2:00