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I am sure this is well known, but I am not an expert...so I appreciate any help

Let $C \subset \mathbb{P}^3$ be the complete intersection of two hypersurfaces of degree $d_1$ and $d_2$. Let $U_{d_1,d_2}$ be the open loci, in the appropriate Hilbert scheme, that parametrize those curves. Hartshorne (Serre?) established a correspondence between this Hilbert scheme $U_{d_1,d_2}$ and the moduli space $M$ of some vector bundles of rank two in $\mathbb{P}^3$.

$\textbf{Question 1}$: Is this true? Where can I read about it?

This implies a compactification of $U_{d_1,d_2}$ will give us a compactification of the moduli space $M$ of those vector bundles.

$\textbf{Question 2}$: Is this a standard procedure for constructing meaningful compactifications of $M$ ?

$\textbf{Question 3}$: What is the situation for elliptic quartic curves ?

Thanks!

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The Serre construction indeed associates to a curve $C$ in $\mathbb{P}^3$ satisfying certain conditions a vector bundle $E$ on $\mathbb{P}^3$ and a global section $s$ of $E$ such that $C$ is the zero set of $s$. But this is trivial for complete intersections: the vector bundle $E$ is just $\mathcal{O}(d_1)\oplus \mathcal{O}(d_2)$, with the section $(F_1,F_2)$, where $F_1=F_2=0$ are te equations of your curve.

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    $\begingroup$ I would add that since the vector bundle is not stable there is no moduli space to consider. Also I would suggest looking into Okonek--Schneider--Spindler book. $\endgroup$
    – Sasha
    Commented Jan 10, 2014 at 11:09

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