Varieties with nef anticanonical divisors and Albanese morphisms of relative dimension one in positive characteristic

Let $X$ be a smooth projective variety with a nef anticanonical divisor over an algebraically closed field of characteristic $p>0$. In this paper, we establish a precise structure of $X$ under the condition that $a_X:X\to \mathrm{A}\mathrm{l}\mathrm{b}(X)$ is of relative dimension one.