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arXiv · 2507.12623

Remarks on two problems by Hassett

Klaus Hulek·Yota Maeda
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Abstract

One of the ultimate goals of the Hassett-Keel program is the determination of the log canonical models of the moduli spaces of pointed rational curves $\overline{M}_{0,n}$. In this paper, we study log canonical models of $\overline{M}_{0,5}$ with \textit{asymmetric} boundary divisors. Our results generalize previous work by Alexeev-Swinarski, Fedorchuk-Smyth, Kiem-Moon and Simpson for the first non-trivial case, namely $n=5$. We prove that all moduli spaces of weighted pointed rational curves $\overline{M}_{0,A}$ arise as log canonical models of $\overline{M}_{0,5}$ for suitable choices of boundary coefficients, thereby also recovering a theorem of Fedorchuk and Moon. In addition, we relate these moduli spaces to Deligne-Mostow ball quotients. We further study log canonical models of the moduli spaces $\overline{M}_{0,n\cdot (1/k)}$ with symmetric weight, which differ from $\overline{M}_{0,n}$. The case $n=5$ can be viewed as an explicit guiding example in a very general program and the paper can thus also serve as an expository introduction.

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Klaus Hulek, Yota Maeda. 2026-04-16. Remarks on two problems by Hassett. https://arxiv.org/abs/2507.12623

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