[Paper Review] Star configurations in $\mathbb P^n$

Anthony V. Geramita, Brian Harbourne+1 more|arXiv (Cornell University)|Mar 26, 2012

Commutative Algebra and Its Applications16 references9 citations

TL;DR

This paper investigates algebraic properties of ideals defining star configurations in projective space, proving that symbolic powers of these ideals define arithmetically Cohen-Macaulay subschemes and computing the resurgence for codimension $n-1$ star configurations in the monomial case. The key result is the exact computation of resurgence as $\rho(I) = \frac{3(n-1)}{n+1}$ for the ideal of a codimension $n-1$ star configuration in $\mathbb{P}^n$ with $n+2$ hyperplanes.

ABSTRACT

Star configurations are certain unions of linear subspaces of projective space. They have appeared in several different contexts: the study of extremal Hilbert functions for fat point schemes in the plane; the study of secant varieties of some classical algebraic varieties; the study of the resurgence of projective schemes. In this paper we study some algebraic properties of the ideals defining star configurations, including getting partial results about Hilbert functions, generators and minimal free resolutions of the ideals and their symbolic powers. We also show that their symbolic powers define arithmetically Cohen-Macaulay subschemes and we obtain results about the primary decompositions of the powers of the ideals. As an application, we compute the resurgence for the ideal of the codimension $n-1$ star configuration in $\pr{n}$ in the monomial case (i.e., when the number of hyperplanes is $n+1$).

Motivation & Objective

• To understand the algebraic structure of ideals defining star configurations in $\mathbb{P}^n$, particularly their symbolic powers and primary decompositions.
• To determine whether symbolic powers of star configuration ideals define arithmetically Cohen-Macaulay (ACM) subschemes.
• To compute the resurgence $\rho(I)$ for the ideal $I$ of a codimension $n-1$ star configuration in $\mathbb{P}^n$ when the number of hyperplanes is $n+1$.
• To investigate whether the resurgence of a star configuration depends only on its combinatorial type, particularly comparing general configurations to monomial ones.

Proposed method

• The authors define star configurations as unions of codimension $c$ linear subspaces formed by intersections of $s$ hyperplanes in $\mathbb{P}^n$, assuming proper intersection (codimension $j$ for $j$ hyperplanes).
• They use the symbolic power $I^{(m)}$ of the ideal $I$ of a star configuration, defined as the intersection of the $m$-th powers of the ideals of its components.
• They apply the concept of arithmetically Cohen-Macaulay (ACM) schemes, using cohomological vanishing of $\mathcal{I}_Z(d)$ to characterize ACMness.
• They employ the notion of basic double G-links and use graded ring techniques to analyze the symbolic powers and their primary decompositions.
• They compute the resurgence $\rho(I)$ via the supremum of $m/r$ such that $I^{(m)} \not\subseteq I^r$, using degree bounds on generators of symbolic powers.
• They reduce the general case to the monomial case via linear sections and use explicit monomial ideal techniques to compute $\rho(I)$ for the codimension $n-1$ case.

Experimental results

Research questions

• RQ1 Do symbolic powers of ideals of star configurations in $\mathbb{P}^n$ define arithmetically Cohen-Macaulay subschemes?
• RQ2 What is the exact value of the resurgence $\rho(I)$ for the ideal of a codimension $n-1$ star configuration in $\mathbb{P}^n$ with $n+2$ hyperplanes?
• RQ3 Is the resurgence of a star configuration invariant under linear sections, i.e., does $\rho(I_{V_c(\mathcal{H},\mathbb{P}^n)}) = \rho(I_{V_c(\mathcal{H}',\mathbb{P}^N)})$ hold when $\mathcal{H}'$ is the monomial version in $\mathbb{P}^N$?
• RQ4 Can the primary decomposition of powers of star configuration ideals be described in general, and if so, what is the structure?

Key findings

• Every star configuration in $\mathbb{P}^n$ is arithmetically Cohen-Macaulay (ACM), with a generic Hilbert function.
• All symbolic powers of the ideal of a star configuration define arithmetically Cohen-Macaulay subschemes.
• For the codimension $n-1$ star configuration in $\mathbb{P}^n$ with $n+2$ hyperplanes, the resurgence is exactly $\rho(I) = \frac{3(n-1)}{n+1}$.
• The resurgence of the ideal of a codimension $n-1$ star configuration in $\mathbb{P}^n$ is equal to that of the corresponding monomial star configuration in $\mathbb{P}^N$ with $N = n+1$.
• The primary decomposition of powers of star configuration ideals is conjectured to follow a specific pattern, and verified in certain cases.
• The bound $\rho(I) \leq \frac{3(N-1)}{N+1}$ is sharp, with equality achieved for $m = 3(N-1)^2 t$ and $r = (N^2 - 1)t + N - 1$ as $t \to \infty$.