[Paper Review] The Minimal Model Program for the Hilbert Scheme of Points on P^2 and Bridgeland Stability

Daniele Arcara, Aaron Bertram+2 more|arXiv (Cornell University)|Mar 1, 2012

Algebraic Geometry and Number Theory20 references16 citations

TL;DR

This paper establishes a precise correspondence between wall-crossings in Bridgeland stability conditions and flips in the minimal model program (MMP) for the Hilbert scheme of points on the projective plane, $π^{2[n]}$. It shows that birational models of $π^{2[n]}$ arising from the MMP are modular: they parametrize Bridgeland semistable objects, which are constructed as GIT quotients of quiver representations, and explicitly computes the stable base locus decomposition and walls for $n \leq 9$. The key contribution is a complete dictionary between Bridgeland stability walls and Mori cone wall-crossings for $n \leq 9$, with explicit descriptions of flips and divisorial contractions.

ABSTRACT

In this paper, we study the birational geometry of the Hilbert scheme of n points on P^2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n at most 9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions.

Motivation & Objective

• To understand the birational geometry of the Hilbert scheme $\mathbb{P}^{2[n]}$ via the minimal model program (MMP).
• To establish a correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings in the Mori cone of $\mathbb{P}^{2[n]}$.
• To provide modular interpretations of the birational models of $\mathbb{P}^{2[n]}$ as moduli spaces of Bridgeland semistable objects.
• To construct these moduli spaces as GIT quotients of quiver representations, proving their projectivity.
• To explicitly compute the stable base locus decomposition and walls for $n \leq 9$, including flips and divisorial contractions.

Proposed method

• The authors run the minimal model program (MMP) on $\mathbb{P}^{2[n]}$, showing it is a Mori dream space with finite stable base locus decomposition.
• They define effective divisors $D_E(n)$ parameterizing subschemes failing to impose independent conditions on sections of a Steiner bundle $E$ on $\mathbb{P}^2$, which span walls in the stable base locus decomposition.
• They use Bridgeland stability conditions on $D^b(\text{coh} \, \mathbb{P}^2)$, identifying walls as semi-circles in the $(s,t)$-plane with centers at $x = -\frac{19}{2}, -\frac{17}{2}, \dots, -\frac{9}{2}$ for $n=9$, corresponding to destabilizing objects like $\mathcal{O}_{\mathbb{P}^2}(-1)$, $\mathcal{I}_p(-1)$, and $\mathcal{O}_{\mathbb{P}^2}(-3)$.
• They construct moduli spaces of Bridgeland semistable objects as GIT quotients of quiver representations, proving these spaces are projective.
• For $n \leq 9$, they explicitly compute the walls in both the Bridgeland stability manifold and the Mori cone, showing a one-to-one correspondence between the two wall sets.
• They verify that the stable base locus in the final chamber for $n=9$ is the locus of schemes failing to impose independent conditions on cubic curves, excluding complete intersections of two cubics, which remain in the stable base locus of the final chamber.

Experimental results

Research questions

• RQ1 How do wall-crossings in the Bridgeland stability manifold for $D^b(\text{coh} \, \mathbb{P}^2)$ correspond to flips and divisorial contractions in the minimal model program for $\mathbb{P}^{2[n]}$?
• RQ2 What is the precise structure of the stable base locus decomposition of the effective cone of $\mathbb{P}^{2[n]}$ for $n \leq 9$?
• RQ3 Can the birational models of $\mathbb{P}^{2[n]}$ obtained via the MMP be given a modular interpretation as moduli spaces of Bridgeland semistable objects?
• RQ4 How can these moduli spaces be constructed explicitly as GIT quotients of quiver representations to ensure projectivity?
• RQ5 What is the stable base locus in the final chamber of the stable base locus decomposition for $\mathbb{P}^{2[9]}$?

Key findings

• For $n \leq 9$, the stable base locus decomposition of the effective cone of $\mathbb{P}^{2[n]}$ is a finite decomposition into rational polyhedral cones, and the authors explicitly determine all walls for $n \leq 9$.
• The effective cone of $\mathbb{P}^{2[n]}$ is spanned by the boundary divisor $B$ and a divisor $D_E(n)$ when $n = \frac{r(r+1)}{2} + s$ with $s/r$ or $1 - \frac{s+1}{r+2}$ in a set related to the golden ratio and Fibonacci ratios.
• The authors construct moduli spaces of Bridgeland semistable objects as GIT quotients of quiver representations, proving they are projective varieties.
• There is a precise one-to-one correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings in the Mori cone of $\mathbb{P}^{2[n]}$, with walls in both settings parameterized by semi-circles in the $(s,t)$-plane.
• For $n=9$, the stable base locus in the final chamber is the locus of schemes of degree 9 that fail to impose independent conditions on curves of degree 3, excluding complete intersections of two cubics.
• Complete intersections of two cubics do not lie in the stable base locus of the final chamber because they impose independent conditions on sections of the bundles $E_k$, which span rays $H - \frac{k}{6k+2}B$ approaching $H - \frac{1}{6}B$ as $k \to \infty$.