On the moduli space of stable surfaces with $p_g=1$ realizing the minimal volume
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917028180262912 |
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| author | Alexeev, Valery Liu, Wenfei Schütt, Matthias |
| author_facet | Alexeev, Valery Liu, Wenfei Schütt, Matthias |
| contents | Let $M_1$ be the moduli space of the KSBA stable surfaces $X$ of geometric genus $p_g(X)=1$ realizing the minimal possible volume $K_X^2=\frac1{143}$. We show that its reduced part $M_{1,\rm red}$ is a $10$-dimensional projective variety isomorphic to the Baily--Borel compactification $\overline{F}_Λ^{\rm BB}$ of the moduli space of $Λ$-polarized K3 surfaces, where $Λ=II_{1,9}\simeq U\oplus E_8$ is a unimodular lattice of signature $(1,9)$. By a result of Brieskorn, $\overline{F}_Λ^{\rm BB}$ is a weighted projective space. We also verify the Viehweg hyperbolicity of the base of a Whitney equisingular family of stable surfaces in $M_1$.
More generally, we prove that the same results hold for the moduli space $M_c$ of KSBA stable pairs $(X,B)$ with coefficients of $B$ belonging to a set $\mathcal C\subset [0,1]$ such that $\mathcal C\cup\{1\}$ attains a minimum, say $c$, and with $p_g(X)=1$, realizing the minimal possible volume $(K_X+B)^2=v(c)$. Indeed, we show that $M_{c,\rm red}$ is independent of $c$ and that for $c\le\frac7{13}$ $M_c$ is isomorphic to $\overline{F}_Λ^{\rm BB}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17678 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the moduli space of stable surfaces with $p_g=1$ realizing the minimal volume Alexeev, Valery Liu, Wenfei Schütt, Matthias Algebraic Geometry 14D22, 14J29, 14J28 Let $M_1$ be the moduli space of the KSBA stable surfaces $X$ of geometric genus $p_g(X)=1$ realizing the minimal possible volume $K_X^2=\frac1{143}$. We show that its reduced part $M_{1,\rm red}$ is a $10$-dimensional projective variety isomorphic to the Baily--Borel compactification $\overline{F}_Λ^{\rm BB}$ of the moduli space of $Λ$-polarized K3 surfaces, where $Λ=II_{1,9}\simeq U\oplus E_8$ is a unimodular lattice of signature $(1,9)$. By a result of Brieskorn, $\overline{F}_Λ^{\rm BB}$ is a weighted projective space. We also verify the Viehweg hyperbolicity of the base of a Whitney equisingular family of stable surfaces in $M_1$. More generally, we prove that the same results hold for the moduli space $M_c$ of KSBA stable pairs $(X,B)$ with coefficients of $B$ belonging to a set $\mathcal C\subset [0,1]$ such that $\mathcal C\cup\{1\}$ attains a minimum, say $c$, and with $p_g(X)=1$, realizing the minimal possible volume $(K_X+B)^2=v(c)$. Indeed, we show that $M_{c,\rm red}$ is independent of $c$ and that for $c\le\frac7{13}$ $M_c$ is isomorphic to $\overline{F}_Λ^{\rm BB}$. |
| title | On the moduli space of stable surfaces with $p_g=1$ realizing the minimal volume |
| topic | Algebraic Geometry 14D22, 14J29, 14J28 |
| url | https://arxiv.org/abs/2510.17678 |