Arakelov inequalities for fibered surfaces in positive characteristic

Let $f:S\to C$ be a relatively minimal semistable fibration of genus $g\ge 2$ over an algebraically closed field of characteristic $p>0$ with smooth and geometrically connected generic fiber. Let $\mathrm{\Sigma }\subset C$ denote the set of points over which the fibers are singular, with $s=\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }$. If the total surface $S$ is Hodge Witt and $s\ge 2$, we prove the classical Arakelov inequality $$\text{deg}{f}_{\ast }{\omega }_{S\mathrm{/}C}\le \frac{g}{2}\text{deg}{\mathrm{\Omega }}_C^1(\log \mathrm{\Sigma }),$$ where ${\omega }_{S\mathrm{/}C}$ denotes the relative dualizing sheaf. For $C={\mathbb{P}}^1$, we obtain the stronger estimate $$\text{deg}{f}_{\ast }{\omega }_{S\mathrm{/}{\mathbb{P}}^1}\le \frac{g}{2}(s-2)-\frac{1}{2}{b}_1(S),$$ where $b_1(S)$ is the first Betti number of $S$. We also construct a genus-$2$ semistable fibration over ${\mathbb{P}}^1$ with exactly $4$ singular fibers in characteristic $5$. Its total surface is Hodge Witt, and $$\text{deg}{f}_{\ast }{\omega }_{S\mathrm{/}{\mathbb{P}}^1}=2=\frac{g}{2}(s-2)\mathrm{.}$$ Thus Nguyen’s lower bound $s\ge 4$ and the Hodge–Witt Arakelov inequality are both sharp. As applications, we derive a canonical-class inequality and a Szpiro-type inequality with coefficients linear in $g$.