[Paper Review] Some Examples of Gorenstein Liaison in Codimension Three

Robin Hartshorne|ArXiv.org|Mar 22, 2001

Algebraic structures and combinatorial models22 references44 citations

TL;DR

This paper investigates Gorenstein liaison in codimension three by constructing examples of points in ℙ³ and curves in ℙ⁴ to test whether all arithmetically Cohen-Macaulay (ACM) schemes are glicci (Gorenstein liaison equivalent to a complete intersection). It shows that while sets of up to 19 points in ℙ³ are glicci, a set of 20 points may not be, and a general (11,7) curve in ℙ⁴ with Rao module k likely lies outside the Gorenstein liaison class of two skew lines, suggesting potential counterexamples to broader conjectures in higher codimension liaison theory.

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.

Motivation & Objective

• To test whether all arithmetically Cohen-Macaulay (ACM) schemes in codimension ≥3 are glicci, i.e., linked via Gorenstein schemes to a complete intersection.
• To investigate the reach of ascending Gorenstein biliaison in linking ACM schemes to simpler ones, particularly in ℙ³ and ℙ⁴.
• To identify potential counterexamples to the conjecture that all ACM schemes are glicci, especially for high-degree configurations.
• To analyze the structure of curves with Rao module k in ℙ⁴, focusing on their liaison class and accessibility via ascending Gorenstein biliaisons.
• To determine whether general ACM curves of degree 20 and genus 26 in ℙ⁴ can be obtained via ascending Gorenstein biliaison from a line, or are not glicci at all.

Proposed method

• Constructing examples of points in ℙ³ lying on planes or quadric surfaces and showing they are glicci via sequences of ascending Gorenstein biliaisons.
• Using the Bordiga surface in ℙ⁴ to parametrize curves of degree 11 and genus 7, computing the dimension of the linear system |C| via Riemann–Roch and cohomological estimates.
• Applying semicontinuity and deformation theory to show that a general (11,7) curve in ℙ⁴ has Rao module k in degree 2, placing it in the family ℳ₂.
• Analyzing the self-intersection C² of curves on the Bordiga surface to bound the dimension of |C| and show that general (11,7) curves do not lie on such surfaces.
• Using the exact sequence 0 → 𝒪_S → 𝒪_S(C) → 𝒪_C(C) → 0 to compute h⁰(𝒪_C(C)) and estimate the number of curves obtainable via the construction.
• Evaluating the cohomology h¹(ℐ_C(n)) for curves in ℙ⁴ to determine the Rao module and assess their liaison behavior.

Experimental results

Research questions

• RQ1 Is every set of 20 points in general position in ℙ³ glicci, or does it serve as a counterexample to the conjecture that all ACM schemes are glicci?
• RQ2 Can a general ACM curve of degree 20 and genus 26 in ℙ⁴ be obtained by ascending Gorenstein biliaison from a line?
• RQ3 Is the general (11,7) curve in ℙ⁴ with Rao module k in the Gorenstein liaison class of two skew lines, or is it not accessible via ascending Gorenstein biliaisons from a minimal curve?
• RQ4 Do minimal curves of every degree ≥2 exist in ℙ⁴ with Rao module k, and can all such curves be reached via ascending Gorenstein biliaisons from a minimal curve?
• RQ5 Are there curves with Rao module k that lie in the Gorenstein liaison class of two skew lines but cannot be obtained by ascending Gorenstein biliaisons from a minimal curve?

Key findings

• Any set of n ≤ 19 points in general position in ℙ³ is glicci, as they can be linked via ascending Gorenstein biliaisons from a single point.
• A set of 20 points in general position in ℙ³ is not known to be glicci, making it a candidate counterexample to the conjecture that all ACM schemes are glicci.
• All ACM curves of degree ≤9 or degree 10 and genus 6 in ℙ⁴ are glicci, and so are certain classes of determinantal curves.
• A general (11,7) curve in ℙ⁴ has Rao module k in degree 2 and does not lie on any Bordiga surface, implying it cannot be obtained by ascending Gorenstein biliaisons from a line.
• The general (11,7) curve in ℙ⁴ is not expected to be in the Gorenstein liaison class of two skew lines, nor reachable via ascending Gorenstein biliaisons from a minimal curve, suggesting it may not be glicci.
• Example 4.6 exhibits a smooth (10,6) curve with Rao module k that is in the Gorenstein liaison class of a minimal curve but cannot be reached by ascending Gorenstein biliaisons from a minimal curve.