An explicit wall crossing for the moduli space of hyperplane arrangements

TL;DR

The paper explicates a wall-crossing phenomenon for moduli spaces of weighted stable hyperplane arrangements, moving from the toric Losev–Manin space $\\overline{\mathrm{LM}}(\\mathbb{P}^d,n)$ to a non-toric moduli $\\overline{\mathrm{M}}_{\\mathbf{nt}}(\\mathbb{P}^d,n)$ by varying weights from $\\mathbf{t}$ to $\\mathbf{nt}$. It proves that there is a birational map $\\overline{\mathrm{M}}_{\\mathbf{nt}}(\\mathbb{P}^d,n) \\rightarrow \\overline{\mathrm{LM}}(\\mathbb{P}^d,n)$, and in the planar case $d=2$ the normalization of the modified space equals the blow-up at the identity, $\\mathrm{Bl}_e\\overline{\mathrm{LM}}(\\mathbb{P}^2,n)$. The results hinge on a careful wall-crossing analysis, stable replacements along one-parameter degenerations, and an extension argument that yields a finite morphism extending the birational map across the exceptional divisor. A key consequence is that for large $n$ any $\\mathbb{Q}$-factorialization of $\\mathrm{Bl}_e\\overline{\mathrm{LM}}(\\mathbb{P}^d,n)$ is not a Mori dream space, highlighting fundamental constraints on the birational geometry of these moduli spaces. For lines in the plane, the authors provide a precise description of the wall-crossing and the induced stable degenerations, clarifying the 2D geometry explicit in the construction.

Abstract

The moduli space of hyperplanes in projective space has a family of geometric and modular compactifications that parametrize stable hyperplane arrangements with respect to a weight vector. Among these, there is a toric compactification that generalizes the Losev-Manin moduli space of points on the line. We study the first natural wall crossing that modifies this compactification into a non-toric one by varying the weights. As an application of our work, we show that any $\mathbb{Q}$-factorialization of the blow up at the identity of the torus of the generalized Losev-Manin space is not a Mori dream space for a sufficiently high number of hyperplanes. Additionally, for lines in the plane, we provide a precise description of the wall crossing.

Figures (5)

• Figure 1: Stable line arrangement for the weights $\mathbf{t}=(1,1,1,\epsilon,\ldots,\epsilon)$ parametrized by the point $e\in\overline{\mathrm{LM}}(\mathbb{P}^2,n)$.
• Figure 2: Crossing of the wall $x_{d+2}+\ldots+x_n=1$ (pictured as the vertical line) moving from $\mathbf{t}$ to $\mathbf{nt}$ in the weight domain $\mathcal{D}(d+1,n)$. For the weight vectors $\mathbf{a},\widehat{\mathbf{w}}$, and $\mathbf{h}$, see Remark \ref{['rmk:idea-behind-the-additional-weights']}.
• Figure 3: The reducible surface $Y$ with the broken lines $C_1,\ldots,C_n$ described in Definition \ref{['def:description-of-the-new-fibers']}.
• Figure 4: For $d=2$, semistable replacement in the proof of Theorem \ref{['thm:locally-at-e-is-simple-blow-up']} before the contraction. The components $E_1,\ldots,E_s$ are isomorphic to $\mathbb{F}_1$.
• Figure 5: In the proof of Lemma \ref{['lem:lc-2-dim']}, line arrangements $(\mathbb{P}^2,\sum_{i=1}^3H_{i,j}+\epsilon\sum_{i=4}^nH_{i,j})$ corresponding to $(X_j,D_j+\sum_{i=1}^3C_i^{(j)}+\epsilon\sum_{i=4}^nC_i^{(j)})$. The solid lines represent $H_{1,j},H_{2,j},H_{3,j}$ and the dotted lines are $H_{4,j},\ldots,H_{n,j}$.

Theorems & Definitions (43)

• Theorem 1.1: Theorem \ref{['thm:locally-at-e-is-simple-blow-up']} and Theorem \ref{['thm:geometric-meaning-blow-up-identity-dim2']}
• Theorem 1.2: Theorem \ref{['thm:Q-fact-not-MDS']}
• Theorem 2.1: Ale08
• Definition 2.2
• Theorem 2.3: Ale08
• Definition 2.4
• Theorem 2.5: Ale15
• Lemma 3.1
• proof
• Definition 3.2