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Kodaira-type classification of singular fibers of some minimal abelian fibrations

Yoon-Joo Kim

Abstract

Let $X \to S$ be a minimal abelian fibration of relative dimension $n$ over a curve. We classify all possible singular fibers $X_s$ having $(n-1)$-dimensional ``abelian variety parts''. This generalizes Kodaira's work on elliptic fibrations, and Matsushita and Hwang--Oguiso's work on Lagrangian fibrations into a single framework. The classification is divided into three parts: semistable, unstable, and multiple. Multiple fibers are again divided into three types: semistable-like, mixed, and unstable-like.

Kodaira-type classification of singular fibers of some minimal abelian fibrations

Abstract

Let be a minimal abelian fibration of relative dimension over a curve. We classify all possible singular fibers having -dimensional ``abelian variety parts''. This generalizes Kodaira's work on elliptic fibrations, and Matsushita and Hwang--Oguiso's work on Lagrangian fibrations into a single framework. The classification is divided into three parts: semistable, unstable, and multiple. Multiple fibers are again divided into three types: semistable-like, mixed, and unstable-like.
Paper Structure (48 sections, 123 theorems, 164 equations, 7 tables)

This paper contains 48 sections, 123 theorems, 164 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Strategy of the proof
  3. Models and delta-regularity
  4. Minimal models and t-automorphism schemes
  5. Delta-regularity
  6. Multiplicity and semistability
  7. Dual of the t-automorphism scheme
  8. Weierstrass models
  9. Irreducible components of the central fiber
  10. Irreducible component after normalization
  11. Irreducible component without normalization
  12. Integral fibers: unstable case
  13. Descent along normalization
  14. Explicit calculations for singular curves
  15. Integral fibers: semistable case
  16. ...and 33 more sections

Key Result

theorem 1.0.1

Let $S$ be a Dedekind scheme of characteristic $0$ with algebraically closed residue fields and $f : X \longrightarrow S$ a minimal $\delta$-regular abelian fibration of relative dimension $n \ge 2$. Write $X_s = f^{-1}(s)$ for a closed fiber. Then

Theorems & Definitions (287)

  • theorem 1.0.1
  • definition 1.0.2
  • theorem 1.0.3: \ref{['thm:splitting']} and \ref{['thm:classification of component group']}
  • definition 2.0.1
  • definition 2.1.1
  • definition 2.1.2
  • remark 2.1.3
  • theorem 2.1.4: kol25:neron
  • theorem 2.1.5: kol25:neron
  • proposition 2.1.6
  • ...and 277 more