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On some boundary divisors in the moduli spaces of stable Horikawa surfaces with $K^2=2p_g-3$

Ciro Ciliberto, Rita Pardini

TL;DR

The paper analyzes the KSBA boundary for Horikawa surfaces with $K^2=2p_g-3$ by focusing on stable surfaces whose only non-canonical singularity is a T-singularity $\frac{1}{4\delta}(1,2\delta-1)$. It establishes that for large $p_g$, the locus $\mathcal D_n$ of such surfaces forms a generically Cartier divisor inside the KSBA moduli, with a general point having a $\frac{1}{4}(1,1)$ singularity and with $\mathcal D_n$ intersecting all components of the Gieseker locus; furthermore, the KSBA moduli space is smooth at a general point of $\mathcal D_n$ and the number of components of $\mathcal D_n$ depends on $p_g\bmod 4$. The work combines explicit degenerations via double covers of Hirzebruch surfaces, deformation theory of $\mathbb Q$-Gorenstein deformations, and a detailed analysis of Horikawa surfaces (first and second kinds) to describe the boundary stratification, the associated divisors, and their local structure. The results illuminate how boundary divisors sit inside KSBA moduli, contribute to the global geometry of the moduli space, and connect to the existing Gieseker strata for Horikawa surfaces. The analysis yields a precise divisorial boundary picture and establishes smoothability properties for general boundary members, with implications for the topology and birational geometry of Horikawa moduli.

Abstract

We describe the normal stable surfaces with K^2=2p_g-3 and p_g>14 whose only non canonical singularity is a cyclic quotient singularity of type 1/4k(1,2k-1) and the corresponding locus D inside the KSBA moduli space of stable surfaces. More precisely, we show that: (1) a general point of any irreducible component of D corresponds to a surface with a singularity of type 1/4(1,1), (2) the closure of D is a divisor contained in the closure of the Gieseker moduli space of canonical models of surfaces with K^2=2p_g-3 and intersects all the components of such closure, and (3) the KSBA moduli space is smooth at a general point of D. In addition, we show that D has 1 or 2 irreducible components, depending on the residue class of p_g modulo 4.

On some boundary divisors in the moduli spaces of stable Horikawa surfaces with $K^2=2p_g-3$

TL;DR

The paper analyzes the KSBA boundary for Horikawa surfaces with by focusing on stable surfaces whose only non-canonical singularity is a T-singularity . It establishes that for large , the locus of such surfaces forms a generically Cartier divisor inside the KSBA moduli, with a general point having a singularity and with intersecting all components of the Gieseker locus; furthermore, the KSBA moduli space is smooth at a general point of and the number of components of depends on . The work combines explicit degenerations via double covers of Hirzebruch surfaces, deformation theory of -Gorenstein deformations, and a detailed analysis of Horikawa surfaces (first and second kinds) to describe the boundary stratification, the associated divisors, and their local structure. The results illuminate how boundary divisors sit inside KSBA moduli, contribute to the global geometry of the moduli space, and connect to the existing Gieseker strata for Horikawa surfaces. The analysis yields a precise divisorial boundary picture and establishes smoothability properties for general boundary members, with implications for the topology and birational geometry of Horikawa moduli.

Abstract

We describe the normal stable surfaces with K^2=2p_g-3 and p_g>14 whose only non canonical singularity is a cyclic quotient singularity of type 1/4k(1,2k-1) and the corresponding locus D inside the KSBA moduli space of stable surfaces. More precisely, we show that: (1) a general point of any irreducible component of D corresponds to a surface with a singularity of type 1/4(1,1), (2) the closure of D is a divisor contained in the closure of the Gieseker moduli space of canonical models of surfaces with K^2=2p_g-3 and intersects all the components of such closure, and (3) the KSBA moduli space is smooth at a general point of D. In addition, we show that D has 1 or 2 irreducible components, depending on the residue class of p_g modulo 4.

Paper Structure

This paper contains 24 sections, 31 theorems, 25 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Moduli spaces and T--singularities
  3. KSBA moduli spaces
  4. T--singularities
  5. Stable 2--Gorenstein T--singular Horikawa surfaces of the second kind
  6. First order deformations of T--singular stable surfaces
  7. Horikawa surfaces
  8. Horikawa surfaces of the first kind
  9. Horikawa surfaces of the second kind
  10. Minimal surfaces with $K^2=2p_g-4$ containing a 2--Gorenstein T--chain
  11. Case (L) of Lemma \ref{['lem:-4iota']}
  12. Cases (L$^\prime$), (Q), (Q$^\prime$) of Lemma \ref{['lem:-4iota']}
  13. Smoothability and the canonical involution
  14. Minimal surfaces with $K^2=2p_g-3$ with a 2-Gorenstein T--singularity
  15. Proof of Theorem \ref{['thm:Dn']}
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $f\colon X\to Z$ be a flat proper morphism with $Z$ reduced and irreducible, such that all the fibers $X_z$, $z\in Z$, are stable surfaces. If the function $z\mapsto K_{X_z}^2$, $z\in Z$, is constant, then $f$ is a stable family.

Theorems & Definitions (63)

  • Theorem 2.1: kollar-families, Thm. 5.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 53 more