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

Abstract

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 $a_X:X\to \mathrm{Alb}(X)$ is of relative dimension one.