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Moduli of elliptic surfaces of Kodaira dimension one fibered over rational curves

Dori Bejleri, Josiah Foster, Andres Fernandez Herrero, Giovanni Inchiostro, Svetlana Makarova, Junyan Zhao

TL;DR

The paper develops a comprehensive framework to understand KSBA moduli spaces for surfaces of Kodaira dimension one, focusing on elliptic surfaces over rational curves. It constructs infinite families of KSBA components via Weierstrass elliptic fibrations and analyzes their boundaries through twisted stable maps and a decorated graph calculus of sliced/stable pruned trees. A key achievement is an explicit wall-crossing description that relates KSBA and KSBA-like compactifications, with detailed results at n=3 and a general obstruction analysis for n>3. The work yields a precise combinatorial stratification of boundary components and provides tools to compute KSBA-stable limits explicitly, illuminating the degenerations to pseudo-elliptic surfaces with controlled singularities.

Abstract

In this article, we construct an infinite sequence of irreducible components of Kollár--Shepherd-Barron (KSB-) moduli spaces of surfaces of arbitrarily large volumes, and describe the boundary of each component completely. Moreover, we describe the stable reduction steps in finding the KSB-limits in an explicit combinatorial way. Our main approach is to study the moduli spaces of elliptic surfaces with Kodaira dimension one, fibered over rational curves, using the techniques of wall-crossing for KSBA moduli and twisted stable maps.

Moduli of elliptic surfaces of Kodaira dimension one fibered over rational curves

TL;DR

The paper develops a comprehensive framework to understand KSBA moduli spaces for surfaces of Kodaira dimension one, focusing on elliptic surfaces over rational curves. It constructs infinite families of KSBA components via Weierstrass elliptic fibrations and analyzes their boundaries through twisted stable maps and a decorated graph calculus of sliced/stable pruned trees. A key achievement is an explicit wall-crossing description that relates KSBA and KSBA-like compactifications, with detailed results at n=3 and a general obstruction analysis for n>3. The work yields a precise combinatorial stratification of boundary components and provides tools to compute KSBA-stable limits explicitly, illuminating the degenerations to pseudo-elliptic surfaces with controlled singularities.

Abstract

In this article, we construct an infinite sequence of irreducible components of Kollár--Shepherd-Barron (KSB-) moduli spaces of surfaces of arbitrarily large volumes, and describe the boundary of each component completely. Moreover, we describe the stable reduction steps in finding the KSB-limits in an explicit combinatorial way. Our main approach is to study the moduli spaces of elliptic surfaces with Kodaira dimension one, fibered over rational curves, using the techniques of wall-crossing for KSBA moduli and twisted stable maps.
Paper Structure (22 sections, 49 theorems, 140 equations, 2 figures, 1 table)

This paper contains 22 sections, 49 theorems, 140 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on KSBA-stable pairs
  3. Background definitions
  4. Canonical models
  5. KSB-stable families
  6. Elliptic surfaces
  7. Minimal Weierstrass fibrations
  8. Twisted fibers
  9. Stability for Weierstrass fibrations
  10. Moduli spaces of elliptic surfaces
  11. Weierstrass fibrations
  12. Locus of elliptic surfaces in the KSBA moduli
  13. Locus of pseudo-elliptic surfaces in the KSB moduli
  14. Local study of the contraction morphism Psin
  15. The case when n=3
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $n\geq 3$ be an integer with $n\neq 4$, and let $0<\epsilon\ll1$ be a rational number. Set $v:=\frac{(n-2)^2}{n}$, $c(\epsilon):=\frac{n-2}{n}-\epsilon$, and $v(\epsilon):=v - n\epsilon$. Then the following statements hold. In particular, the following are all birational to each other, for each $n\geq 3$, $n\neq 4$: the moduli stack $\mathcal{K}_n$, the moduli stack $\mathcal{W}_n^{\mathop{\m

Figures (2)

  • Figure 1: Special fibers $\mathscr{X}^{(i)}_{0}\dashrightarrow \mathscr{X}^{(i+1)}_{0}$ of a flip of La Nave
  • Figure 2: Notations as in the proof of \ref{['prop_pruning_tree']}.

Theorems & Definitions (139)

  • Theorem 1.1: = \ref{['proposition: Phi defines iso of W with KSBA']} + \ref{['prop: existence of the morphism']} + \ref{['thm_proof_psi_n_isom']} + \ref{['thm: irreducibility of K_n']} + \ref{['proposition: maps between compactifications']}
  • Theorem 1.2: =\ref{['algorith: stable reduction']} + \ref{['prop_pruning_tree']} + \ref{['thm:stratificationE_n']}
  • Theorem 1.3: =\ref{['cor_amm_moduli_spaces_are_isom']} + \ref{['thm:boundary_P_n']}
  • Corollary 1.4
  • Theorem 1.5: =\ref{['cor_KSB_limits_have_at_most_6_vertices']} + \ref{['prop_K3_smooth_and_of_dim_28']} + \ref{['thm:embedding_3']}
  • Remark 1.6
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: KM98*Def. 2.11
  • Definition 2.6: kollar_modbook*Def. 11.5
  • ...and 129 more