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On canonical bundle formula for fibrations of curves with arithmetic genus one

Jingshan Chen, Chongning Wang, Lei Zhang

TL;DR

We develop canonical bundle formulas for fibrations of relative dimension one over fields of characteristic $p>0$, addressing both separable and inseparable cases. In the separable setting we obtain a Witaszek-type decomposition after finite purely inseparable base changes, with coefficients and an effective divisor governing the relationship between $D$ and the base/total-space canonical data. For inseparable fibrations, especially when the base $S$ has maximal Albanese dimension, we derive lower bounds on the Kodaira dimension of $D$ and show that, under mild singularities, the Albanese map $a_X$ becomes a fibration. Applications include structural results for varieties with nef anticanonical divisor and a detailed base-change framework for genus-one curves over imperfect fields, enabling a robust analysis of the canonical bundle in positive characteristic.

Abstract

In this paper, we develop canonical bundle formulas for fibrations of relative dimension one in characteristic $p>0$. For such a fibration from a log pair $f\colon (X, Δ) \to S$, if $f$ is separable, we can obtain a formula similar to the one due to Witaszek \cite{Wit21}; if $f$ is inseparable, we treat the case when $S$ is of maximal Albanese dimension. As an application, we prove that for a klt pair $(X,Δ)$ with $-(K_X+Δ)$ nef, if the Albanese morphism $a_X\colon X \to A$ is of relative dimension one, then $X$ is a fiber space over $A$.

On canonical bundle formula for fibrations of curves with arithmetic genus one

TL;DR

We develop canonical bundle formulas for fibrations of relative dimension one over fields of characteristic , addressing both separable and inseparable cases. In the separable setting we obtain a Witaszek-type decomposition after finite purely inseparable base changes, with coefficients and an effective divisor governing the relationship between and the base/total-space canonical data. For inseparable fibrations, especially when the base has maximal Albanese dimension, we derive lower bounds on the Kodaira dimension of and show that, under mild singularities, the Albanese map becomes a fibration. Applications include structural results for varieties with nef anticanonical divisor and a detailed base-change framework for genus-one curves over imperfect fields, enabling a robust analysis of the canonical bundle in positive characteristic.

Abstract

In this paper, we develop canonical bundle formulas for fibrations of relative dimension one in characteristic . For such a fibration from a log pair , if is separable, we can obtain a formula similar to the one due to Witaszek \cite{Wit21}; if is inseparable, we treat the case when is of maximal Albanese dimension. As an application, we prove that for a klt pair with nef, if the Albanese morphism is of relative dimension one, then is a fiber space over .
Paper Structure (11 sections, 19 theorems, 24 equations)

This paper contains 11 sections, 19 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The behavior of the canonical divisor under quotient of foliations and purely inseparable base changes
  4. Foliations and purely inseparable morphisms
  5. "Pushing-down" foliations along a fibration
  6. Foliations associated with morphisms arising from base changes
  7. The movable part and fixed part
  8. The behavior of the relative canonical divisors under base changes
  9. Curves of Arithmetic genus one
  10. Auxiliary results
  11. Notation and assumptions

Key Result

Theorem 1.1

Let $(X, \Delta)$ be a projective log canonical pair defined over an algebraically closed field $k$ of characteristic $p > 0$, and let $f\colon X \to S$ be a fibration of relative dimension one such that $K_X + \Delta \sim_{\mathbb{Q}} f^*D$ for some $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $D$ on for some rational number $t\in [0,1]$ and an effective $\mathbb{Q}$-divisor $\Delta_T$ on $T$.

Theorems & Definitions (33)

  • Theorem 1.1: Wit21*Theorem 3.4
  • Theorem 1.2: see Theorem \ref{['thm:sep-cb-formula']}
  • Theorem 1.3: see Subsection \ref{['sec:proof-cbf']}
  • Remark 1.4
  • Theorem 1.5: see Theorem \ref{['thm:S-abelian']}
  • Lemma 2.1: Zha19*Lemma 4.2
  • Lemma 2.2
  • proof
  • Remark
  • Lemma 2.4
  • ...and 23 more