Tropical methods for stable octic double planes

Technical Summary: Tropical Methods for Stable Octic Double Planes

Problem Statement
The paper addresses the classification of the Kollár–Shepherd-Barron–Alexeev (KSBA) moduli space of surfaces of general type with fixed Chern numbers $K^2=2$ and $p_g=3$. These smooth surfaces are octic double planes (double covers of $\mathbb{P}^2$ branched over a smooth octic curve). While the KSBA compactification includes singular "stable" surfaces, the boundary strata corresponding to non-Gorenstein singularities have been largely uncharted. Previous work by Anthes classified the Gorenstein limits and provided initial examples of non-Gorenstein limits. This paper aims to systematically classify all normal KSBA-stable surfaces in this moduli space that admit a $\mathbb{Q}$-Gorenstein smoothing to a smooth octic double plane, with a specific focus on the non-Gorenstein case.

Methodology
The authors employ techniques from tropical geometry and mirror symmetry, specifically the framework developed by Gross, Hacking, Keel, and Kontsevich (GHK), to organize the case analysis of surface degenerations.

1. Reduction to Manetti Surfaces: The authors establish that any normal KSBA limit of an octic double plane is a double cover of a degeneration of $\mathbb{P}^2$. Citing work by Bădescu, Manetti, Hacking, and Prokhorov, they restrict the base surfaces to "Manetti surfaces"—degenerations of $\mathbb{P}^2$ with at worst quotient singularities (specifically Wahl singularities and simple elliptic singularities). They prove that simple elliptic singularities cannot occur in the base of a log canonical octic double cover, leaving only Manetti surfaces with Wahl singularities.
2. Mirror Tropicalisation: The geometry of these Manetti surfaces is encoded via "mirror tropicalisations" (integral affine manifolds). The authors utilize the correspondence between Manetti surfaces and Markov triples $(a, b, c)$ satisfying $a^2+b^2+c^2=3abc$. Toric Manetti surfaces correspond to weighted projective planes $\mathbb{P}(a^2, b^2, c^2)$, while non-toric ones (denoted $HP(S)$) correspond to partial smoothings of these.
3. Enumeration of Sections: The branch curves of the double covers correspond to sections of a specific line bundle. In the tropical picture, these sections are in bijection with integer points within the mirror tropicalisation polygon. The authors enumerate these integer points for all relevant Markov triples.
4. Discrepancy Calculation: Using the tropical data, the authors compute the discrepancies of the singularities of the pair $(Y, \frac{1}{2}B)$, where $Y$ is the base Manetti surface and $B$ is the branch curve. A necessary condition for the double cover to be KSBA-stable is that the pair must be log canonical (discrepancies $\geq -1$).
5. Mutation and Induction: The authors use "mutations" of the tropical polygons (analogous to mutations of Markov triples) to inductively rule out infinitely many cases. They demonstrate that for most Manetti surfaces, the branch curve necessarily acquires a multiple component or singularities that violate the log canonical condition.

Key Contributions and Results

• Classification Theorem: The paper provides a complete classification of normal KSBA-stable surfaces with $K^2=2, p_g=3$ admitting a $\mathbb{Q}$-Gorenstein smoothing. The stable surfaces fall into four categories:

  1. Double covers of $\mathbb{P}^2$ branched over singular octics (Gorenstein limits, previously classified by Anthes).
  2. Double covers of $Y = \mathbb{P}(1, 1, 4)$ branched over a curve of weighted degree 16. These may possess two $\frac{1}{4}(1,1)$ singularities or specific baskets of singularities arising from $\mathbb{Z}/2$-quotients of simple elliptic or cusp singularities.
  3. Double covers of $Y = \mathbb{P}(1, 4, 25)$ branched over a curve of weighted degree 80. These possess a $\frac{1}{50}(1, 29)$ singularity and one of the baskets listed for $\mathbb{P}(1, 1, 4)$.
  4. Double covers of the non-toric Manetti surface $HP(5)$ (a partial smoothing of $\mathbb{P}(1, 4, 25)$). These possess a $\frac{1}{50}(1, 29)$ singularity.

• Exclusion of Higher Markov Numbers: A significant result is the proof that Manetti surfaces associated with Markov numbers $c \notin \{1, 2, 5, 13, 29, 34\}$ (and their associated non-toric variants) do not yield log canonical double covers. The authors show that for these surfaces, the branch curve must contain a component with multiplicity $\geq 2$ or the resulting singularities fail the log canonical condition.

• Detailed Analysis of Singularities: The paper provides explicit descriptions of the minimal resolutions for the non-Gorenstein cases.

  • For $Y=\mathbb{P}(1, 1, 4)$, the authors analyze cases where the branch curve passes through the singularity. They identify dual graphs for the minimal resolutions, which include $\mathbb{Z}/2$-quotients of simple elliptic singularities and cusp singularities.
  • They establish bounds on the parameters of these singularities (e.g., for the cusp case, $t \leq 19$ and $t_1 + t_2 \leq 38$, with a conjecture that $t_1 + t_2 \leq 21$).
  • They explicitly construct the $\frac{1}{50}(1, 29)$ Wahl singularity arising in the $\mathbb{P}(1, 4, 25)$ and $HP(5)$ cases.

• Moduli Space Structure: The authors calculate the dimensions of the strata in the KSBA boundary corresponding to these singular surfaces. They show that the strata are nested (e.g., $D^{IIc}_Y \subset D^{IIb}_Y \subset D^{IIa}_Y \subset D^I_Y$) and determine their codimensions relative to the 36-dimensional moduli space of smooth octic double planes.

Significance and Claims
The paper claims that tropical geometry and mirror symmetry provide a powerful, systematic framework for organizing the complex case analysis required to study KSBA moduli spaces near the Noether line ($2p_g = K^2 + 4$). By translating the problem of classifying surface degenerations into the combinatorial problem of counting integer points in tropical polygons and analyzing their mutations, the authors are able to tame an otherwise intractable classification problem.

The work extends the understanding of KSBA-stable surfaces beyond the Gorenstein case, providing the first comprehensive list of non-Gorenstein limits for octic double planes. The authors note that while their techniques are motivated by the study of Horikawa surfaces (which lie on or near the Noether line), the specific classification of the $K^2=2, p_g=3$ case serves as a foundational step. They explicitly state that they have not studied all combinations of Gorenstein and non-Gorenstein singularities, nor have they determined which specific combinations of the identified baskets are realizable, though they prove the existence of the strata. The paper also highlights that while the identified surfaces are KSBA-stable, most are not smoothable in the sense of having vanishing local-to-global obstruction groups, though they can be rationally blown down to smooth 4-manifolds.