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

Valery Alexeev, Wenfei Liu, Matthias Schütt

TL;DR

The paper analyzes the moduli of KSBA stable surfaces with $p_g=1$ realizing the minimal volume $K_X^2=\frac{1}{143}$ and extends the study to stable surface pairs with boundary coefficients from a capped set $\mathcal C$. It proves that the reduced moduli $M_{1,\mathrm{red}}$ is a 10-dimensional projective variety isomorphic to the BB compactification $\overline{F}_\Lambda^{\mathrm{BB}}$ for $\Lambda=II_{1,9}$, which Brieskorn shows is a weighted projective space; the approach relies on degenerations of K3 surfaces and lattice-polarized K3 moduli. A parallel statement holds for moduli $M_c$ with $p_g(X)=1$ and minimal volume $v(c)$, where $M_{c,\mathrm{red}}$ is independent of $c$ and $M_c\cong\overline{F}_\Lambda^{\mathrm{BB}}$ for $c\le\tfrac{7}{13}$. The paper also computes log canonical rings in these minimal cases, provides two proofs of the $M_c\simeq \overline{F}_\Lambda^{\mathrm{BB}}$ identification (via degenerations and via Brieskorn’s explicit family), and confirms Viehweg hyperbolicity for Whitney equisingular families in $M_c$ with maximal variation. The results connect KSBA moduli with lattice-polarized K3 moduli, yielding explicit descriptions of compactifications and degeneration behavior relevant for moduli theory in higher dimensions.

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 $K_X^2=\frac1{143}$. 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 $\overline{F}_Λ^{\rm BB}$.

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

TL;DR

The paper analyzes the moduli of KSBA stable surfaces with realizing the minimal volume and extends the study to stable surface pairs with boundary coefficients from a capped set . It proves that the reduced moduli is a 10-dimensional projective variety isomorphic to the BB compactification for , which Brieskorn shows is a weighted projective space; the approach relies on degenerations of K3 surfaces and lattice-polarized K3 moduli. A parallel statement holds for moduli with and minimal volume , where is independent of and for . The paper also computes log canonical rings in these minimal cases, provides two proofs of the identification (via degenerations and via Brieskorn’s explicit family), and confirms Viehweg hyperbolicity for Whitney equisingular families in with maximal variation. The results connect KSBA moduli with lattice-polarized K3 moduli, yielding explicit descriptions of compactifications and degeneration behavior relevant for moduli theory in higher dimensions.

Abstract

Let be the moduli space of the KSBA stable surfaces of geometric genus realizing the minimal possible volume . We show that its reduced part is a -dimensional projective variety isomorphic to the Baily--Borel compactification of the moduli space of -polarized K3 surfaces, where is a unimodular lattice of signature . By a result of Brieskorn, is a weighted projective space. We also verify the Viehweg hyperbolicity of the base of a Whitney equisingular family of stable surfaces in . More generally, we prove that the same results hold for the moduli space of KSBA stable pairs with coefficients of belonging to a set such that attains a minimum, say , and with , realizing the minimal possible volume . Indeed, we show that is independent of and that for is isomorphic to .
Paper Structure (11 sections, 21 theorems, 61 equations, 2 figures)

This paper contains 11 sections, 21 theorems, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stable surfaces with , achieving the minimal volume
  4. Log canonical rings of stable surfaces
  5. Proof of Theorem \ref{['thm: main1-ii']} using degenerations of K3 surfaces
  6. Generalities on moduli of (quasi)polarized K3 surfaces
  7. Lattice-polarized K3 surfaces for $\Lambda=U\oplus E_8$
  8. Compactifications of $F_\Lambda$ and their maps to $M_c$
  9. The recognizability of $R$
  10. Proof of Theorem \ref{['thm: main1-ii']} using Brieskorn's family
  11. Hyperbolicity of the bases of equisingular families in

Key Result

Theorem 1.1

Let $M_c$ be the moduli space of stable surface pairs $(X, B)\in \mathcal{S}(\mathcal{C},1)$ such that $(K_X+B)^2=v(c)$. Then the isomorphism class of its reduced part $M_{c,\rm red}$ does not depend on the parameter $c$.

Figures (2)

  • Figure 1: Divisor model for $(S_0,\epsilon R_0^\mathrm{rc})$ and the modified pair $(S_0, \epsilon R_0)$
  • Figure 2: M1 modifications achieving a divisor model for $(S_0,\epsilon R_0)$

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 3.2: liu2017minimal-volume
  • proof
  • Definition 3.3
  • Definition 3.4
  • ...and 38 more