We show that a weak version of the canonical bundle formula holds for fibrations of relative dimension one. We provide various applications thereof, for instance, using the recent result of Xu and Zhang, we prove the log non-vanishing conjecture for three-dimensional klt pairs over any algebraically closed field k of characteristic \(p>5\), which in turn implies the termination of any sequence of three-dimensional flips in the pseudo-effective case. We also show the log abundance conjecture for threefolds over k when the nef dimension is not maximal, and the base point free theorem for threefolds over \(\mathbb {{\overline{F}}}_p\) when \(p>2\).

我们表明，规范丛公式的弱版本适用于相对维数为 1 的纤维化。我们提供了它的各种应用，例如，使用 Xu 和 Zhang 的最新结果，我们证明了对具有特征\(p>5\) 的任何代数闭域k 上的三维 klt 对的对数非消失猜想，其中turn 意味着在伪有效情况下终止任何三维翻转序列。我们还展示了当 nef 维度不是最大时k 上三倍的对数丰度猜想，以及\( \(p>2 )时\( \mathbb {{\overline{F}}}}_p\) 上三倍的基点自由定理\)。