Multiple Structures with Arbitrarily Large Projective Dimension on Linear Subspaces

Abstract

Let $K$ be an algebraically closed field. There has been much interest in characterizing multiple structures in ${\P }_K^n$ 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 $(h,e)\ne (2,2)$ 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.

21 pages (fixed typo in statement of main theorem from version 1)