Multiple structures with arbitrarily large projective dimension supported on linear subspaces

Let K   be an algebraically closed field. There has been much interest in characterizing multiple structures in PKn 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 unmixedness. Specifically, we prove that for any positive integers h,e≥2h,e≥2 with (h,e)≠(2,2)(h,e)≠(2,2) and p≥5p≥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/IR/I is e  , (3) the projective dimension of R/IR/I is at least p and (4) the ideal I   is primary to a linear prime (x1,…,xh)(x1,…,xh). 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.