  Let $S$ be a polynomial ring over an algebraic closed field $k$ and $
\mathfrak p =(x,y,z,w) $ a homogeneous height four prime ideal. We give a
finite characterization of the degree two component of ideals primary to
$\mathfrak p$, with multiplicity $e \leq 3$. We use this result to give a tight
bound on the projective dimension of almost complete intersections generated by
five quadrics with $e \leq 3$.

Commutative Algebra