Minimal Homogenous Liaison and Licci Ideals

TL;DR

This paper investigates the properties of homogeneously licci ideals in polynomial rings, demonstrating that not all such ideals can be linked to complete intersections using minimal degree forms, and showing limitations of Hilbert functions and Betti numbers in classifying these ideals.

Contribution

It provides counterexamples to the minimal homogeneously licci property and shows that Hilbert functions and Betti numbers do not distinguish licci classes.

Findings

Counterexamples for n ≥ 28 points in P^3 that are homogeneously licci but not minimally homogeneously licci

Hilbert functions cannot distinguish between homogeneously licci and non-licci ideals

Betti numbers cannot distinguish between homogeneously licci and minimally homogeneously licci ideals

Abstract

We study the linkage classes of homogeneous ideals in polynomial rings. An ideal is said to be homogeneously licci if it can be linked to a complete intersection using only homogeneous regular sequences at each step. We ask a natural question: if $I$ is homogeneously licci, then can it be linked to a complete intersection by linking using regular sequences of forms of smallest possible degree at each step (we call such ideals minimally homogeneously licci)? In this paper we answer this question in the negative. In particular, for every $n\geq 28$ we construct a set of $n$ points in $\mathbb P^3$ which are homogeneously licci, but not minimally homogeneously licci. Moreover, we prove that one cannot distinguish between the classes of homogeneously licci and non-licci ideals based only on their Hilbert functions, nor distinguish between homogeneously licci and minimally homogeneously licci ideals based solely on the graded Betti numbers. Finally, by taking hypersurface sections, we show that the natural question has a negative answer whenever the height of the ideal is at least three.