Multiple Structures with Arbitrarily Large Projective Dimension on Linear Subspaces

TL;DR

The paper demonstrates that in algebraic geometry, there is no finite way to characterize multiple structures on linear subspaces with only Serre's (S_1) property, by constructing ideals with arbitrarily large projective dimension.

Contribution

It shows the impossibility of finite characterization of such structures under minimal assumptions, contrasting with previous results for Cohen-Macaulay cases.

Findings

Constructs ideals with arbitrary large projective dimension

Shows no finite classification exists under Serre's (S_1) property

Contrasts with known classifications for Cohen-Macaulay structures

Abstract

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.