Geometry of the stability scattering diagram for $\mathbb{P}^2$ and applications

Abstract

We give a detailed analysis of the stability scattering diagram for $\mathbb{P}^2$ introduced by Bousseau. This scattering diagram lives in a subset of $\mathbb{R}^2$, and we decompose this subset into three regions, $R_{\Delta},R_{\Diamond}$ and $R_{\mathrm{unbdd}}$. The region $R_{\Delta}$ has a chamber structure whose chambers are in one-to-one correspondence with strong exceptional triples. No ray of the stability scattering diagram enters the interior of such a triangle, replicating a result of Prince, and generalizing a result of Bousseau. The region $R_{\Diamond}$ is decomposed into diamonds, which are in one-to-one correspondence with exceptional bundles. Each diamond has a vertical diagonal corresponding to a rank zero object and is traversed by a dense set of rays. Crucially, however, there are no collisions of rays inside diamonds, making it still possible to control the scattering diagram in $R_{\Diamond}$. Finally, the behaviour of $R_{\mathrm{unbdd}}$ is chaotic, in that every rational point inside it is a collision of an infinite number of rays. We show that the bounded region $R_{\mathrm{bdd}}= R_{\Delta}\cup R_{\Diamond}$ has as upper boundary the Le Potier curve, thus showing that this curve arises naturally through the algorithmic scattering process. We give an application of these results by describing the first wall-crossing for the moduli space of one-dimensional rank zero objects on $\mathbb{P}^2$. In the sequel, we apply these results to describe the full Bridgeland wall-crossing for $\mathrm{Hilb}^n(\mathbb{P}^2)$ for any $n$.