[Paper Review] Containment results for ideals of various configurations of points in P^N

Cristiano Bocci, Susan Cooper+1 more|arXiv (Cornell University)|Sep 9, 2011

Commutative Algebra and Its Applications15 references14 citations

TL;DR

This paper proposes and verifies a new conjecture (Conjecture 3.9) on symbolic power containments for ideals of points in projective space, extending recent conjectures by Harbourne and Huneke. It proves the conjecture for various configurations of points in $\mathbb{P}^N$, including generic and special point arrangements, using bounds on symbolic and ordinary powers via Seshadri constants and regularity estimates.

ABSTRACT

Guided by evidence coming from a few key examples and attempting to unify previous work of Chudnovsky, Esnault-Viehweg, Eisenbud-Mazur, Ein-Lazarsfeld-Smith, Hochster-Huneke and Bocci-Harbourne, Harbourne and Huneke recently formulated a series of conjectures that relate symbolic and regular powers of ideals of fat points in ${\bf P}^N$. In this paper we propose another conjecture along the same lines (Conjecture 3.9), and we verify it and the conjectures of Harbourne and Huneke for a variety of configurations of points.

Motivation & Objective

• To extend and unify recent conjectures relating symbolic and ordinary powers of ideals of fat points in $\mathbb{P}^N$.
• To propose a new conjecture (Conjecture 3.9) on containment of symbolic powers in products of maximal ideal powers and ordinary powers.
• To verify the new conjecture and existing conjectures (e.g., Harbourne-Huneke) for various configurations of points in $\mathbb{P}^N$.
• To strengthen known containment results using Seshadri constants and Castelnuovo-Mumford regularity in characteristic 0.
• To provide theoretical and computational evidence for the conjectures through analysis of generic and special point sets, including line count configurations.

Proposed method

• Propose a new containment conjecture (Conjecture 3.9) of the form $I^{(t(m+1))} \subseteq M^t (I^{(m)})^t$ for ideals of points in $\mathbb{P}^N$.
• Use Seshadri constants $\varepsilon(n) \geq 1/\sqrt{n+1}$ to bound $\alpha(I^{(m)})$ and $\operatorname{reg}(I^{(m)})$ for generic points.
• Apply regularity bounds: $\operatorname{reg}(I^{(m)}) \leq (m+1)\sqrt{n+1} - 2$ for $n > 9$ generic points in $\mathbb{P}^2$.
• Leverage Lemma 2.6, which states that if $\alpha(J) > t \cdot \operatorname{reg}(I) + t$, then $J \subseteq M^t I^t$.
• Use the inequality $\alpha(I^{(r)}) \geq \alpha(I^{(m)}) + r - m$ (Lemma 2.4) to compare symbolic power degrees.
• Apply known results: $I^{(Nt + (m-1)t)} \subseteq (I^{(m)})^t$ and $I^{(Nt + (m-1)t + s)} \subseteq M^s (I^{(m)})^t$ in characteristic 0.

Experimental results

Research questions

• RQ1 Does the new conjecture $I^{(t(m+1))} \subseteq M^t (I^{(m)})^t$ hold for ideals of generic points in $\mathbb{P}^2$?
• RQ2 Can the Harbourne-Huneke conjectures be verified for configurations of points beyond known examples?
• RQ3 What is the sharp bound on $\alpha(I^{(m)})$ for ideals of $n$ generic points in $\mathbb{P}^2$?
• RQ4 How do Seshadri constants and Castelnuovo-Mumford regularity constrain symbolic power containments?
• RQ5 Under what conditions does $I^{(t(N+m-1)+s)} \subseteq M^s (I^{(m)})^t$ hold in characteristic 0?

Key findings

• Conjecture 3.9 is verified for $n \geq 10$ generic points in $\mathbb{P}^2$ when $1 \leq m \leq \sqrt{n+1} - 1$, via Seshadri constant and regularity bounds.
• For $n \geq 16$ generic points, $I^{(t(m+1))} \subseteq M^t (I^{(m)})^t$ holds due to $\alpha(I^{(t(m+1))}) \geq t(m+1)\sqrt{n}$ and $\operatorname{reg}(I^{(m)}) \leq (m+1)\sqrt{n+1} - 2$.
• In characteristic 0, $I^{(t(N+m-1)+s)} \subseteq M^s (I^{(m)})^t$ holds for all $s,t,m \geq 1$, strengthening known containment results.
• The bound $\alpha(I^{(m)}) \geq m$ is sufficient to ensure $\alpha(I^{(m)})/m \geq 1$, supporting the conjectural lower bound $\alpha(I^{(m)})/m \geq (\alpha(I) + N - 1)/N$.
• For $n > 9$ not a square, $\operatorname{reg}(I^{(m)}) \leq (m+1)\sqrt{n+1} - 2$ holds, enabling the verification of containment via Lemma 2.6.
• Conjecture 3.5 is verified for line count configurations in $\mathbb{P}^2$ in characteristic 0, based on separate work by the second author.