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

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 $f\text{\hspace{0.17em}⁣}:(X,\mathrm{\Delta })\to S$, 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 $(X,\mathrm{\Delta })$ with $-(K_X+\mathrm{\Delta })$ nef, if the Albanese morphism $a_X\text{\hspace{0.17em}⁣}:X\to A$ is of relative dimension one, then $X$ is a fiber space over $A$.