[Paper Review] Multiplicity of Codimension Three Almost Complete Intersections

Sumi Seo, Hema Srinivasan|ArXiv.org|Dec 5, 2007

Polynomial and algebraic computation9 references4 citations

TL;DR

This paper establishes the upper bound of the multiplicity conjecture for codimension three almost complete intersection rings via direct computation using the Buchsbaum-Eisenbud structure theorem for Gorenstein ideals. It proves that the multiplicity of such rings is bounded above by the product of maximal shifts divided by 3!, confirming the conjecture in this case and providing partial results on the lower bound under specific conditions.

ABSTRACT

We establish the upper bound in the multiplicity conjecture of Herzog, Huneke and Srinivasan for the codimension three almost complete intersections. We also give some partial results in the case where I is the aci linked to a complete intersection in one step.

Motivation & Objective

• To establish the upper bound in the Herzog-Srinivasan multiplicity conjecture for codimension three almost complete intersections.
• To analyze the structure of minimal free resolutions of such ideals using the duality of mapping cones.
• To investigate conditions under which the lower bound of the conjecture also holds.
• To explore the relationship between almost complete intersections and linked Gorenstein ideals.
• To provide explicit formulas for multiplicity in terms of Betti number shifts and generator degrees.

Proposed method

• Utilizes the Buchsbaum-Eisenbud structure theorem to represent codimension three almost complete intersections as (K:J), where J is a Gorenstein ideal and K is a regular sequence of three elements.
• Constructs the minimal free resolution of R/I as the dual of the mapping cone of the resolutions of R/J and R/K.
• Employs the homogeneous resolution of R/J via Pfaffians of a skew-symmetric matrix with homogeneous entries.
• Derives explicit formulas for the maximal and minimal shifts M_i and m_i in the resolution of R/I in terms of degrees of generators of K and J.
• Applies the formula e(R/I) = ∑β_{i,j}(-1)^i j to compute multiplicity from Betti numbers.
• Uses inequalities on degrees and sums of d_i to prove bounds under specific conditions.

Experimental results

Research questions

• RQ1 Does the upper bound of the Herzog-Srinivasan multiplicity conjecture hold for codimension three almost complete intersections?
• RQ2 Under what conditions does the lower bound of the multiplicity conjecture also hold for such ideals?
• RQ3 How do the Betti number shifts M_i and m_i relate to the degrees of generators of the defining ideal and its linked Gorenstein ideal?
• RQ4 Can the multiplicity of R/I be bounded solely in terms of the maximal shifts M_i and the codimension?
• RQ5 What structural properties emerge when the regular sequence linking I and J is generated in a single degree?

Key findings

• The upper bound of the multiplicity conjecture holds for all codimension three almost complete intersections, with e(R/I) ≤ (M₁M₂M₃)/6.
• The upper bound is proven under the condition ∑_{i=2}^3 (e_i - e₁) ≥ d₁, where e_i are degrees of the regular sequence and d₁ is the smallest generator degree of the Gorenstein ideal J.
• When the regular sequence is generated in a single degree e, and M₁ = e, the upper bound holds due to M_i = ie for i = 1,…,n−1 and e^{n−1} ≤ n ∏_{i=2}^n d_i.
• If m_{n−1} = (n−1)e, the lower bound ∏ m_i / n! ≤ e(R/J) holds, under the condition e ≥ d_n + d_{n−1}.
• The multiplicity of R/J is given by e(R/J) = e^n − ∏_{i=1}^n d_i, where e is the common degree of the regular sequence and d_i are the degrees of the Gorenstein generators.
• In the case of a regular sequence in degree e, the upper bound holds if e ≥ (1/(n−1))∑_{i=1}^n d_i, which is equivalent to ∑(e_i − e₁) ≥ d₁.