  Let $K$ be an algebraically closed field. There has been much interest in
characterizing multiple structures in $\P^n_K$ defined on a linear subspace of
small codimension under additional assumptions (e.g. Cohen-Macaulay). We show
that no such finite characterization of multiple structures is possible if one
only assumes Serre's $(S_1)$ property holds. Specifically, we prove that for
any positive integers $h, e \ge 2$ with $(h,e) \neq (2,2)$ and $p \ge 5$ there
is a homogeneous ideal $I$ in a polynomial ring over $K$ such that (1) the
height of $I$ is $h$, (2) the Hilbert-Samuel multiplicity of $R/I$ is $e$, (3)
the projective dimension of $R/I$ is at least $p$ and (4) the ideal $I$ is
primary to a linear prime $(x_1,..., x_h)$. This result is in stark contrast to
Manolache's characterization of Cohen-Macaulay multiple structures in
codimension 2 and multiplicity at most 4 and also to Engheta's characterization
of unmixed ideals of height 2 and multiplicity 2.
