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Moduli of surfaces fibered in (log) Calabi-Yau pairs II: elliptic surfaces

Giovanni Inchiostro, Junyan Zhao

TL;DR

This paper advances the KSBA moduli program for surfaces fibered in lc-trivial fibrations by focusing on elliptic surfaces and exploiting a fiberwise involution to pass to a log Calabi–Yau fibration on a quotient Y. It classifies the boundary objects in the KSBA moduli space of elliptic surfaces with a bisection and then recovers the moduli stacks of elliptic surfaces with a section using a streamlined approach that avoids the La Nave flip by leveraging the canonical bundle formula and ISZ25. A central technical thread is the analysis of cyclic covers and double covers, which permits a precise description of fibers over nodal and smooth loci and yields a concrete list of permissible singularities (A_k, D_m, E_7, E_6, E_8 types) in various ramification regimes. The Weierstrass fibration case is treated via a two-step KSBA-stable extension over twisted curves, providing a robust compactification framework for both rational and K3 elliptic surfaces, and yielding practical descriptions of degenerations and moduli, including explicit descriptions of the base curve data and moduli parts. Overall, the work delivers a cohesive framework for compactifying moduli spaces of elliptic surfaces and Weierstrass fibrations, with broad implications for understanding degenerations and stable limits in higher-dimensional moduli theory.

Abstract

This paper continues the study initiated in [ISZ25] on the moduli of surfaces admitting lc-trivial fibrations. Using the techniques developed in [ISZ25], we (1) provide a classification of the surfaces appearing on the boundary of the KSBA-moduli space of elliptic surfaces with a bisection (2) recover the results of a series of papers on the moduli stacks of elliptic surfaces with a section [AB22, Inc20, Bru15]. Notably, our proof of (2) avoids the use of explicit steps of an MMP, such as the "La Nave flip" from [LN02], which plays a central role in [AB22,Inc20].

Moduli of surfaces fibered in (log) Calabi-Yau pairs II: elliptic surfaces

TL;DR

This paper advances the KSBA moduli program for surfaces fibered in lc-trivial fibrations by focusing on elliptic surfaces and exploiting a fiberwise involution to pass to a log Calabi–Yau fibration on a quotient Y. It classifies the boundary objects in the KSBA moduli space of elliptic surfaces with a bisection and then recovers the moduli stacks of elliptic surfaces with a section using a streamlined approach that avoids the La Nave flip by leveraging the canonical bundle formula and ISZ25. A central technical thread is the analysis of cyclic covers and double covers, which permits a precise description of fibers over nodal and smooth loci and yields a concrete list of permissible singularities (A_k, D_m, E_7, E_6, E_8 types) in various ramification regimes. The Weierstrass fibration case is treated via a two-step KSBA-stable extension over twisted curves, providing a robust compactification framework for both rational and K3 elliptic surfaces, and yielding practical descriptions of degenerations and moduli, including explicit descriptions of the base curve data and moduli parts. Overall, the work delivers a cohesive framework for compactifying moduli spaces of elliptic surfaces and Weierstrass fibrations, with broad implications for understanding degenerations and stable limits in higher-dimensional moduli theory.

Abstract

This paper continues the study initiated in [ISZ25] on the moduli of surfaces admitting lc-trivial fibrations. Using the techniques developed in [ISZ25], we (1) provide a classification of the surfaces appearing on the boundary of the KSBA-moduli space of elliptic surfaces with a bisection (2) recover the results of a series of papers on the moduli stacks of elliptic surfaces with a section [AB22, Inc20, Bru15]. Notably, our proof of (2) avoids the use of explicit steps of an MMP, such as the "La Nave flip" from [LN02], which plays a central role in [AB22,Inc20].
Paper Structure (18 sections, 41 theorems, 93 equations, 3 figures)

This paper contains 18 sections, 41 theorems, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. KSBA-moduli stacks for certain ruled surfaces
  3. Moduli of elliptic surfaces with a bisection
  4. Cyclic covers of KSBA-stable pairs
  5. Classification of fibers on double covers: preparation
  6. Classification of fibers over nodal locus of $C$
  7. $\mathcal{L}$ is Cartier
  8. $\mathcal{L}$ is of Cartier index $2$
  9. Singular fibers over the smooth locus: no ramified fibers
  10. Singular fibers over the smooth locus: with ramified fibers
  11. Example: K3 surfaces
  12. Moduli of Weierstrass fibrations
  13. Geometry of Weierstrass fibrations
  14. KSBA-moduli stack of Weierstrass fibrations
  15. Proof of Theorem \ref{['thm_W_fib_with_a_section']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\overline{\mathcal{M}}^{\mathop{\mathrm{\mathrm{KSBA}}}\nolimits}_{\textup{ell},2}(\epsilon_1,\epsilon_2)$ be the closure of the locus of elliptic surfaces as above, and let $(X,\epsilon_1R+\epsilon_2F)$ be a pair appearing on the boundary of this moduli space. Then:

Figures (3)

  • Figure 1: Surfaces of different types
  • Figure 2: Log canonical model morphism $Y\rightarrow Y^{(1)}$. In blue, the curves to which some irreducible components of $Y_0$ are contracted to.
  • Figure 3: The pair $\left({\mathbb P}_B,\ \Sigma_{{\mathbb P}_B} + \Sigma_2 + \Delta + F\right)$

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2: cf. Kol23
  • Theorem 2.3: cf. ISZ25
  • Theorem 2.4: cf. ISZ25
  • Remark 2.5
  • Corollary 2.6
  • proof
  • Corollary 2.7
  • ...and 62 more