Remarks on two problems by Hassett
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913037634502656 |
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| author | Hulek, Klaus Maeda, Yota |
| author_facet | Hulek, Klaus Maeda, Yota |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_12623 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Remarks on two problems by Hassett Hulek, Klaus Maeda, Yota Algebraic Geometry Number Theory 14H10, 14D20, 11F03, 14E30 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. |
| title | Remarks on two problems by Hassett |
| topic | Algebraic Geometry Number Theory 14H10, 14D20, 11F03, 14E30 |
| url | https://arxiv.org/abs/2507.12623 |