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Explicit KSBA compactifications of moduli spaces of secondary and tertiary Burniat surfaces

Valery Alexeev, Xiaoyan Hu

TL;DR

This work constructs explicit KSBA compactifications for the moduli spaces of Burniat surfaces of degrees $5$, $4$, and $3$ by compactifying via slc stable pairs and tracking their ${ m Z}_2^2$-cover structures. It extends the prior primary Burniat (degree $6$) results by detailing toric degenerations, boundary strata, and the relabeling symmetries that act on RGB data, yielding precise stack and coarse moduli descriptions. The method combines toric geometry, abelian covers, and MMP-based degenerations to enumerate irreducible components and boundary divisors, including explicit descriptions of the toric and non-toric degenerations and the necessary flips. The findings provide a complete KSBA picture for all four secondary/tertiary types, with applications to related families and potential extensions to similar line-configurations-based constructions.

Abstract

We describe explicitly the geometric KSBA compactifications, obtained by adding slc surfaces~$X$ with ample canonical class, of moduli spaces of Burniat surfaces of degrees $K^2=5$, $4$ and $3$.

Explicit KSBA compactifications of moduli spaces of secondary and tertiary Burniat surfaces

TL;DR

This work constructs explicit KSBA compactifications for the moduli spaces of Burniat surfaces of degrees , , and by compactifying via slc stable pairs and tracking their -cover structures. It extends the prior primary Burniat (degree ) results by detailing toric degenerations, boundary strata, and the relabeling symmetries that act on RGB data, yielding precise stack and coarse moduli descriptions. The method combines toric geometry, abelian covers, and MMP-based degenerations to enumerate irreducible components and boundary divisors, including explicit descriptions of the toric and non-toric degenerations and the necessary flips. The findings provide a complete KSBA picture for all four secondary/tertiary types, with applications to related families and potential extensions to similar line-configurations-based constructions.

Abstract

We describe explicitly the geometric KSBA compactifications, obtained by adding slc surfaces~ with ample canonical class, of moduli spaces of Burniat surfaces of degrees , and .
Paper Structure (15 sections, 9 theorems, 39 equations, 13 figures, 5 tables)

This paper contains 15 sections, 9 theorems, 39 equations, 13 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Definition of Burniat surfaces
  3. Degree $6$
  4. The main theorem
  5. Degenerations
  6. The underlying surfaces
  7. The toric setup
  8. ${\mathbb Z}_2^2$-covers and the relabeling group
  9. Evolution of irreducible components
  10. Degree $3$
  11. Degree $4$ non-nodal
  12. Degree $4$ nodal
  13. Degree $5$
  14. Figures
  15. Tables

Key Result

Theorem 1.1

The KSBA compactification ${\overline M}^{\rm Bur}_{3}$ is an $S_2$-quotient of ${\mathbb P}^1$ (and is isomorphic to ${\mathbb P}^1$). The boundary ${\overline M}^{\rm Bur}_3\setminus M^{\rm Bur}_3$ consists of two divisors. The underlying moduli stack is a ${\mathbb Z}_2^2$-gerbe over the quotient

Figures (13)

  • Figure 1: Burniat arrangements on ${\mathbb P}^2$ and $\Sigma=\mathop{\mathrm{Bl}}\nolimits_3{\mathbb P}^2$
  • Figure 2: Cases of degree 5, 4 non-nodal, 4 nodal, 3
  • Figure 3: Surfaces underlying the stable pairs $(Y_6,\sum_{i=0}^3\frac{1}{2}(R_i+G_i+B_i))$ in ${\overline M}_6$
  • Figure 4: Irreducible components of $Y_6$ appearing in ${\overline M}_6$, with volumes. The underlying surfaces are: $\texttt{\#}0(6)\ \Sigma\texttt{\#}1(2)\,{\mathbb P}^1\times{\mathbb P}^1\texttt{\#}2(2)\,{\mathbb P}^1\times{\mathbb P}^1\texttt{\#}3(3)\, {\mathbb F}_1\texttt{\#}4(1)\,{\mathbb P}^2\texttt{\#}5(2)\,{\mathbb P}^1\times{\mathbb P}^1\texttt{\#}6(4)\,{\mathbb P}^2\texttt{\#}7(5)\,\mathop{\mathrm{Bl}}\nolimits_1\Sigma\texttt{\#}8(1)\,{\mathbb P}^2\texttt{\#}9(3)\,{\mathbb F}_1$
  • Figure 5: New irreducible components appearing after blowups and contractions by $|mL'|$. The underlying surfaces are: $\texttt{\#}3_1(2)\ {\mathbb P}^1\times{\mathbb P}^1\texttt{\#}6_1(3)\ {\mathbb F}_1\texttt{\#}6_2(2)\ {\mathbb P}^1\times{\mathbb P}^1\texttt{\#}6_3(1)\ {\mathbb P}^2\texttt{\#}9_1(2)\ {\mathbb P}(1,1,2)$
  • ...and 8 more figures

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 3.1
  • Lemma 5.1
  • proof
  • Remark 5.2
  • Lemma 6.1
  • ...and 2 more