Stein degree on non-Fano type fibrations

TL;DR

The paper addresses the unboundedness of the strong Stein degree for vertical divisors on non-Fano type log Calabi-Yau fibrations, contrasting with known boundedness in the Fano-type setting. It builds explicit examples by leveraging genus-one reduction results from LLR04 to produce arithmetic surfaces with split multiplicative reduction, and then applies Artin approximation to spread these examples to finite-type, quasi-projective and eventually projective log Calabi-Yau fibrations. The main contribution is showing that for integers $n$ and $d$ with $d\mid n$ and $d\neq n$, there exists a log Calabi-Yau fibration where a horizontal component has $\operatorname{sdeg}(S^{\operatorname{nor}}/C)=d$, hence $\operatorname{ssdeg}$ is unbounded. This demonstrates that the boundedness phenomena for Stein degrees on Fano-type fibrations do not extend to non-Fano cases, informing moduli considerations and the structure of stable minimal models in this broader context.

Abstract

We construct examples showing that Stein degree of vertical divisors on non-Fano type log Calabi-Yau fibrations is unbounded.

Theorems & Definitions (12)

• Theorem 1.1: B-moduli
• Theorem 1.2: birkar-Q-25
• Theorem 1.3: birkar-Q-25
• Theorem 1.4
• Proposition 3.2: cf. LLR04
• proof
• Theorem 3.4
• proof
• Theorem 3.6
• proof