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Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces

Valery Alexeev, Rita Pardini

TL;DR

The paper provides explicit KSBA compactifications for two components of the moduli space of surfaces of general type: Campedelli surfaces with $igl| ext{pi}_1(X)igr|={f Z}_2^3$ and Burniat surfaces with $K_X^2=6$. It develops a framework combining abelian covers and toric degenerations via fiber fans to describe the compactifications: for Campedelli, the main component is the quotient ${ m GL}(3,{f F}_2)ackslash({f P}^2)^7//{ m PGL}(3)$; for Burniat, the normalization of the compactified moduli is ${ar M}(1/2)/oldsymbol{\\Gamma}$, with a detailed toric degeneration structure encoded by ${ar M}^{ m tor}$ and its fiber fans. The approach yields an explicit, highly structured degeneration analysis, including minimal, maximal, and nontoric degenerations, and connects to weighted-line arrangements via Matroid tilings. These results provide concrete, computable moduli spaces for zero-boundary KSBA compactifications and offer tools for studying boundary strata, deformations, and automorphism actions in these classical surfaces. The methods have potential applications to other abelian-cover examples and to broader KSBA compactifications of higher-dimensional varieties.

Abstract

We describe explicitly the geometric compactifications, obtained by adding slc surfaces $X$ with ample canonical class, for two connected components in the moduli space of surfaces of general type: Campedelli surfaces with $π_1(X)=\mathbb Z_2^3$ and Burniat surfaces with $K^2=6$.

Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces

TL;DR

The paper provides explicit KSBA compactifications for two components of the moduli space of surfaces of general type: Campedelli surfaces with and Burniat surfaces with . It develops a framework combining abelian covers and toric degenerations via fiber fans to describe the compactifications: for Campedelli, the main component is the quotient ; for Burniat, the normalization of the compactified moduli is , with a detailed toric degeneration structure encoded by and its fiber fans. The approach yields an explicit, highly structured degeneration analysis, including minimal, maximal, and nontoric degenerations, and connects to weighted-line arrangements via Matroid tilings. These results provide concrete, computable moduli spaces for zero-boundary KSBA compactifications and offer tools for studying boundary strata, deformations, and automorphism actions in these classical surfaces. The methods have potential applications to other abelian-cover examples and to broader KSBA compactifications of higher-dimensional varieties.

Abstract

We describe explicitly the geometric compactifications, obtained by adding slc surfaces with ample canonical class, for two connected components in the moduli space of surfaces of general type: Campedelli surfaces with and Burniat surfaces with .

Paper Structure

This paper contains 28 sections, 28 theorems, 39 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compact moduli of stable surfaces
  4. Abelian covers
  5. Campedelli surfaces with $\pi_1(X)={\mathbb Z}_2^3$
  6. Definitions
  7. Compact moduli spaces
  8. Proof of Theorem \ref{['thm-intro:campedelli']}
  9. Degenerate Campedelli surfaces and their singularities
  10. Burniat surfaces with $K_X^2=6$
  11. Set-up and notation
  12. Variation of weights
  13. The toric setup
  14. Minimal (codimension $1$) toric degenerations
  15. Maximal (codimension $4$) toric degenerations
  16. ...and 13 more sections

Key Result

Theorem 1

For Campedelli surfaces, the main irreducible component of the compactification ${\overline M}_{\rm Cam}^{\rm slc}$ is a finite $\mathop{\mathrm{GL}}\nolimits(3,{\mathbb F}_2)$-quotient of a smooth projective GIT quotient $({\mathbb P}^2)^7//\mathop{\mathrm{PGL}}\nolimits(3)$.

Figures (12)

  • Figure 1: Burniat arrangements on ${\mathbb P}^2$ and $\Sigma=\mathop{\mathrm{Bl}}\nolimits_3{\mathbb P}^2$
  • Figure 2: Minimal toric degenerations of types A, B, C, D.
  • Figure 3: Degeneration for case A
  • Figure 4: Degeneration for case C
  • Figure 5: Further case C degenerations over $0$ and $\infty\in{\mathbb P}^1$
  • ...and 7 more figures

Theorems & Definitions (79)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: kollar2023families-of-varieties, Thm. 8.1
  • Definition 2.6
  • Example 2.7: ${\mathbb Z}_2^2$-covers
  • ...and 69 more