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Tropical methods for stable octic double planes

Jonathan David Evans, Angelica Simonetti, Giancarlo Urzúa

TL;DR

This work studies the KSBA boundary for octic double planes with $K^2=2$ and $p_g=3$ by combining tropical geometry and mirror symmetry. The authors reduce degenerations to double covers of Manetti surfaces, formulate mirror tropicalisations, and enumerate integer points to control possible octic branch divisors. A finite list of candidate normal KSBA stable octic double Manetti surfaces is obtained, and each case is analyzed via discrepancy computations to determine log canonicity. The results yield a complete classification of normal KSBA-stable degenerations and illuminate how tropical methods organize the complicated case analysis. The approach connects Horikawa-type surfaces to almost toric geometry and offers tools for broader KSBA studies of surfaces on or near the Noether line.

Abstract

This paper has been written to illustrate the power of techniques from tropical geometry and mirror symmetry for studying the KSBA moduli space of surfaces on or near the Noether line. We focus on the moduli space of octic double planes ($K^2 = 2$, $p_g = 3$) and use methods from tropical and toric geometry to classify the strata corresponding to normal KSBA-stable surfaces, focusing on the non-Gorenstein case.

Tropical methods for stable octic double planes

TL;DR

This work studies the KSBA boundary for octic double planes with and by combining tropical geometry and mirror symmetry. The authors reduce degenerations to double covers of Manetti surfaces, formulate mirror tropicalisations, and enumerate integer points to control possible octic branch divisors. A finite list of candidate normal KSBA stable octic double Manetti surfaces is obtained, and each case is analyzed via discrepancy computations to determine log canonicity. The results yield a complete classification of normal KSBA-stable degenerations and illuminate how tropical methods organize the complicated case analysis. The approach connects Horikawa-type surfaces to almost toric geometry and offers tools for broader KSBA studies of surfaces on or near the Noether line.

Abstract

This paper has been written to illustrate the power of techniques from tropical geometry and mirror symmetry for studying the KSBA moduli space of surfaces on or near the Noether line. We focus on the moduli space of octic double planes (, ) and use methods from tropical and toric geometry to classify the strata corresponding to normal KSBA-stable surfaces, focusing on the non-Gorenstein case.
Paper Structure (77 sections, 59 equations, 19 figures, 1 table)

This paper contains 77 sections, 59 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theorem.
  3. Remark.
  4. Remark.
  5. Remark.
  6. Remark.
  7. Remark.
  8. Overview.
  9. Acknowledgements.
  10. Background
  11. Double covers.
  12. Discrepancies of pairs.
  13. Classification of log canonical surface singularities.
  14. Toric geometry.
  15. Almost toric geometry.
  16. ...and 62 more sections

Figures (19)

  • Figure 1: The dual graphs for minimal resolutions of the singularities appearing in Case II of Theorem \ref{['thm:main_thm']}.
  • Figure 2: The dual graphs of the minimal resolutions of the elliptic quotient singularities admitting smoothings with Milnor number zero.
  • Figure 3: A Wahl vertex with index $3$ and eigendirection $(1,1)$.
  • Figure 4: The mutations $\mu_0$ (left) and $\mu_1$ (right).
  • Figure 5: The Markov topograph. Shapes and shading of the vertices will be explained in \ref{['dfn:Z_polygons']}--\ref{['pg:integer_points']}. In the end, we will see that the only normal KSBA-stable octic double Manetti surfaces are double covers of $\mathbb{H}\mathbb{P}(1)$, $\mathbb{H}\mathbb{P}(2)$ and $\mathbb{H}\mathbb{P}(5)$, shown in bold here.
  • ...and 14 more figures

Theorems & Definitions (15)

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  • ...and 5 more