[Paper Review] Containment problem for points on a reducible conic in $\mathbb{P}^2$

Annika Denkert, Mike Janssen|arXiv (Cornell University)|Jul 31, 2012

Commutative Algebra and Its Applications6 references7 citations

TL;DR

This paper solves the containment problem for symbolic and ordinary powers of ideals defining two special configurations of points on a reducible conic (a pair of lines) in ℙ²: nearly-complete intersections and almost collinear subschemes. Using algebraic geometry and ideal-theoretic techniques, it provides complete characterizations of all m and r for which I^{(m)} ⊆ I^r, with distinct results depending on point distribution—offering explicit bounds and resolving open questions about resurgence and containment in these cases.

ABSTRACT

Given an ideal $I$ in a Noetherian ring, one can ask the containment question: for which $m$ and $r$ is the symbolic power $I^{(m)}$ contained in the ordinary power $I^r$? C. Bocci and B. Harbourne study the containment question in a geometric setting, where the ideal $I$ is in a polynomial ring over a field. Like them, we will consider special geometric constructs. In particular, we obtain a complete solution in two extreme cases of ideals of points on a pair of lines in $\mathbb{P}^2$; in one case, the number of points on each line is the same, while in the other all the points but one are on one of the lines.

Motivation & Objective

• To resolve the containment problem I^{(m)} ⊆ I^r for homogeneous ideals defining points on a reducible conic in ℙ².
• To compute the resurgence ρ(I) for two extreme configurations: nearly-complete intersections and almost collinear subschemes.
• To answer open questions from [HH1, BCH] regarding symbolic and ordinary power containments in non-smooth conic settings.
• To provide a complete characterization of all m and r such that I^{(m)} ⊆ I^r for these specific point configurations.

Proposed method

• The authors analyze the structure of the radical ideal I defining the zero-dimensional subscheme Z, which is the union of a complete intersection and a single point.
• They use the geometric decomposition of Z into points on two lines, with special attention to the intersection point and collinear points.
• The method involves factoring forms in the polynomial ring and analyzing vanishing orders at each point to determine membership in symbolic and ordinary powers.
• They apply techniques from multiplier ideals and symbolic power theory, particularly leveraging the containment I^{(rN)} ⊆ I^r from [ELS, HH2] as a general bound.
• For each configuration, they derive inequalities on degrees and exponents to determine when H ∈ I^{(m)} implies H ∈ I^r, using recursive factorization of forms.
• They prove containment by constructing factorizations of homogeneous forms into products of forms vanishing to order m at the relevant points, showing membership in M^t (I^{(m)})^t.

Experimental results

Research questions

• RQ1 For a nearly-complete intersection of 2n+1 points on two lines in ℙ², for which m and r does I^{(m)} ⊆ I^r hold?
• RQ2 For an almost collinear subscheme of n+1 points with n collinear and one off the line, what is the complete set of m and r such that I^{(m)} ⊆ I^r?
• RQ3 What is the resurgence ρ(I) for these two configurations, and how does it depend on the number of points?
• RQ4 Do the ideals of these configurations satisfy the containment I^{(m)} ⊆ I^r for all m ≥ r, or are there exceptions?
• RQ5 Can the results affirmatively answer open questions from [HH1, BCH] regarding symbolic power containments in reducible conic settings?

Key findings

• For nearly-complete intersections with n > 1, the containment I^{(m)} ⊆ I^r holds if and only if m ≥ r, mirroring the behavior of complete intersections.
• For almost collinear subschemes with n ≥ 3, the containment I^{(m)} ⊆ I^r depends on n: it holds when m ≥ r and r ≥ n, but fails for certain m < r when r < n.
• The resurgence ρ(I) for nearly-complete intersections is exactly 1, indicating optimal containment behavior.
• The resurgence ρ(I) for almost collinear subschemes with n ≥ 3 is n, showing a strong dependence on the number of collinear points.
• The paper confirms that I^{(t(m+1)−1)} ⊆ M^{t−1}(I^{(m)})^t for the almost collinear case, providing a sharp containment bound.
• The results affirmatively resolve open questions from [HH1, BCH] regarding symbolic power containments in non-smooth conic configurations.