[Paper Review] A boundary divisor in the moduli space of stable quintic surfaces

Julie Rana|arXiv (Cornell University)|Jul 26, 2014

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper establishes that stable numerical quintic surfaces (with $K^2 = \chi = 5$) having a unique non-Du Val singularity must be either a $\frac{1}{4}(1,1)$ or $\frac{1}{9}(1,2)$ Wahl singularity, with the former being the only possibility under additional Noether line conditions. Using extended deformation theory in the log setting, the authors identify a 39-dimensional boundary divisor in the KSBA compactification $\overline{\mathcal{M}}_{5,5}$, arising from surfaces obtained via double covers of $\mathbb{P}^1 \times \mathbb{P}^1$ or a quadric cone with specific branch loci.

ABSTRACT

We give a bound on which singularities may appear on Kollár--Shepherd-Barron--Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces with $K^2=χ=5$) whose unique non Du Val singularity is a Wahl singularity. We then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space $\overline{\mathcal{M}}_{5,5}$ corresponding to these surfaces. Quintic surfaces are the simplest examples of surfaces of general type and the question of describing their moduli is a long-standing question in algebraic geometry.

Motivation & Objective

• To classify stable numerical quintic surfaces with a unique non-Du Val singularity.
• To determine which Wahl singularities can occur on such surfaces under the constraint $K_W^2 = K_S^2 + 1$, where $S$ is the minimal model of the resolution.
• To describe the boundary divisor in the KSBA compactification $\overline{\mathcal{M}}_{5,5}$ corresponding to these surfaces using extended deformation theory.
• To characterize the moduli space $\mathcal{M}_{5,5}$ by identifying explicit geometric constructions of the stable surfaces via double covers of rational surfaces.

Proposed method

• Applies a refined bound on Wahl singularity length under the condition $K_W^2 = K_S^2 + 1$, proving only lengths 1 or 2 are possible.
• Uses Horikawa’s classification of surfaces on the Noether line to restrict possible singularities when $K_W^2 = 2p_g - 3$, showing only $\frac{1}{4}(1,1)$ singularities occur.
• Extends Horikawa’s deformation theory to the log setting, analyzing the Kuranishi map and Schouten bracket on $H^1(X, T_X(\log C))$ to study obstructions and smoothability.
• Constructs stable surfaces as double covers of $\mathbb{P}^1 \times \mathbb{P}^1$ or a quadric cone, with branch loci intersecting rulings in specified ways (e.g., tangential intersections or nodes).
• Analyzes the Lie bracket on $H^1(X, T_X(\log C)) \otimes H^1(X, T_X(\log C))$ to show surjectivity of the Schouten bracket, implying non-trivial deformation space.
• Uses $\mathbb{Z}/2\mathbb{Z}$-equivariance and invariance properties of cohomology to show $[\rho_1, \rho_i] \neq 0$ for some $i > 1$, confirming non-vanishing of the degree-two part of the Kuranishi map.

Experimental results

Research questions

• RQ1 Which Wahl singularities can appear on a stable numerical quintic surface with $K^2 = \chi = 5$ and a unique non-Du Val singularity?
• RQ2 What is the maximum possible length of a Wahl singularity on such a surface under the condition $K_W^2 = K_S^2 + 1$?
• RQ3 How can the boundary divisor in $\overline{\mathcal{M}}_{5,5}$ be described geometrically and deformation-theoretically?
• RQ4 What is the structure of the Kuranishi map for log tangent sheaves in the context of double covers with prescribed branch loci?
• RQ5 Can the moduli space $\mathcal{M}_{5,5}$ be stratified by geometric constructions involving double covers of rational surfaces with specific intersection types?

Key findings

• Only Wahl singularities of length 1 or 2 can occur on stable numerical quintic surfaces with $K_W^2 = K_S^2 + 1$, corresponding to $\frac{1}{4}(1,1)$ or $\frac{1}{9}(1,2)$ singularities.
• When $K_W^2 = 2p_g - 3$, the unique non-Du Val singularity must be a $\frac{1}{4}(1,1)$ singularity, and if $K_W^2 > 3$, the minimal model $S$ is of general type.
• The boundary divisor in $\overline{\mathcal{M}}_{5,5}$ is 39-dimensional and corresponds to surfaces obtained as double covers of $\mathbb{P}^1 \times \mathbb{P}^1$ or a quadric cone with branch curves intersecting rulings in specified ways.
• The Schouten bracket on $H^1(X, T_X(\log C))$ is surjective, implying the deformation space is non-trivial and the moduli space has a smooth boundary component.
• The Kuranishi map’s degree-two part is non-vanishing due to non-zero $[\rho_1, \rho_i]$ for some $i > 1$, confirming the existence of non-trivial first-order deformations.
• The minimal resolution of a type 1 surface arises from a double cover of $\mathbb{P}^1 \times \mathbb{P}^1$ branched over a sextic tangent to a diagonal at six points, with the preimage of the diagonal being two $(-4)$-curves intersecting at six points.