English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $K$ be an algebraically closed field. There has been much interest incharacterizing multiple structures in $\P^n_K$ defined on a linear subspace ofsmall codimension under additional assumptions (e.g. Cohen-Macaulay). We showthat no such finite characterization of multiple structures is possible if oneonly assumes Serre's $(S_1)$ property holds. Specifically, we prove that forany positive integers $h, e \ge 2$ with $(h,e) \neq (2,2)$ and $p \ge 5$ thereis a homogeneous ideal $I$ in a polynomial ring over $K$ such that (1) theheight 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 $I$ isprimary to a linear prime $(x_1,..., x_h)$. This result is in stark contrast toManolache's characterization of Cohen-Macaulay multiple structures incodimension 2 and multiplicity at most 4 and also to Engheta's characterizationof unmixed ideals of height 2 and multiplicity 2.

Multiple Structures with Arbitrarily Large Projective Dimension on Linear Subspaces