  We study the lowest dimensional open case of the question whether every
arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^N$ is glicci, that is,
whether every zero-scheme in $\mathbb{P}^3$ is glicci. We show that a set of $n
\geq 56$ points in general position in $\PP^3$ admits no strictly descending
Gorenstein liaison or biliaison. In order to prove this theorem, we establish a
number of important results about arithmetically Gorenstein zero-schemes in
$\mathbb{P}^3$.
