Some Examples of Gorenstein Liaison in Codimension Three

TL;DR

This paper explores the extension of Gorenstein liaison theory from codimension 2 to higher codimensions, specifically analyzing points in P^3 and curves in P^4, revealing both promising results and potential counterexamples.

Contribution

It investigates the applicability of codimension 2 liaison results to higher codimensions, providing new examples and identifying limitations of current theories.

Findings

Successful extension for small degree cases

Examples suggesting limitations of the theory

Potential counterexamples to extension hypotheses

Abstract

Gorenstein liaison seems to be the natural notion to generalize to higher codimension the well-known results about liaison of varieties of codimension~2 in projective space. In this paper we study points in ${\mathbb P}^3$ and curves in ${\mathbb P}^4$ in an attempt to see how far typical codimension~2 results will extend. While the results are satisfactory for small degree, we find in each case examples where we cannot decide the outcome. These examples are candidates for counterexamples to the hoped-for extensions of codimension~2 theorems.