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$\newcommand\P{\mathbb P} \newcommand\C{\mathbb C}$For a degree 4 curve, the Macaulay quartic curve is the only irreducible space curve (up to isomorphism) for which it remains an open problem to find two algebraic surfaces in $\P^3(\C)$ whose intersection is this curve. For reducible degree 4 (connected) space curves I would expect it to be simpler, but there is one that I cannot work out: Consider a smooth conic section that connects two skew lines in $\P^3(\C)$ and such that none of these lines lie in the same plane as the conic section. Is it possible to find two algebraic surfaces whose intersection is precisely the union of these curves?

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    $\begingroup$ What have you tried? Are you looking for irreducible surfaces, or would reducible be fine? $\endgroup$
    – Tabes Bridges
    Commented Dec 9 at 6:51

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