An explicit wall crossing for the moduli space of hyperplane arrangements

Patricio Gallardo; Luca Schaffler

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arXiv · 2405.09051

An explicit wall crossing for the moduli space of hyperplane arrangements

Patricio Gallardo·Luca Schaffler

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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.

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Patricio Gallardo, Luca Schaffler. 2024-05-15. An explicit wall crossing for the moduli space of hyperplane arrangements. https://arxiv.org/abs/2405.09051

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