Effective nonvanishing of $K$-trivial regular surfaces over imperfect fields

Source abstract

For any regular projective surface $X$ over an arbitrary field $k$ of characteristic $p>0$ with $K_X\equiv0$, we prove that $$60K_X\sim0 .$$ This uniform bound is optimal. To show that the factor $5$ is necessary, we construct an example in characteristic $5$ for which $K_X$ has exact order five. If $X$ is geometrically normal, then $12K_X\sim0$. Our main focus is the geometrically non-normal case, where we obtain more precise bounds according to the Albanese morphism. If it is nontrivial, then $p=2$ or $3$, and we prove $12K_X\sim0$ and $6K_X\sim0$, respectively; both bounds are optimal. If it is trivial, then $K_X\sim0$ for $p\ge7$, while $5K_X\sim0$, $3K_X\sim0$, and $4K_X\sim0$ for $p=5,3,2$, respectively.