On the moduli space of stable surfaces with $p_g=1$ realizing the minimal volume

Abstract

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 $K_X^2=\frac{1}{143}$. We show that its reduced part $M_{1,\mathrm{red}}$ is a $10$-dimensional projective variety isomorphic to the Baily--Borel compactification ${\overline{F}}_{\Lambda }^{\mathrm{BB}}$ of the moduli space of $\Lambda$-polarized K3 surfaces, where $\Lambda =I{I}_{1,9}\simeq U\oplus E_8$ is a unimodular lattice of signature $(1,9)$. By a result of Brieskorn, ${\overline{F}}_{\Lambda }^{\mathrm{BB}}$ 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 $M_{c,\mathrm{red}}$ is independent of $c$ and that for $c\le \frac{7}{13}$ $M_c$ is isomorphic to ${\overline{F}}_{\Lambda }^{\mathrm{BB}}$.