Arakelov inequalities for fibered surfaces in positive characteristic

Abstract

Let $f:S\to C$ be a relatively minimal semistable fibration of genus $g\ge2$ over an algebraically closed field of characteristic $p>0$ with smooth and geometrically connected generic fiber. Let $\Sigma\subset C$ denote the set of points over which the fibers are singular, with $s=|\Sigma|$. If the total surface $S$ is Hodge Witt and $s\ge2$, we prove the classical Arakelov inequality $$ \text{deg}\, f_*\omega_{S/C}\le \frac g2 \text{deg}\, \Omega_C^1(\log\Sigma), $$ where $\omega_{S/C}$ denotes the relative dualizing sheaf. For $C=\mathbb P^1$, we obtain the stronger estimate $$ \text{deg}\, f_*\omega_{S/\mathbb P^1}\le \frac g2(s-2)-\frac12 b_1(S), $$ 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 $$ \text{deg}\, f_*\omega_{S/\mathbb P^1}=2=\frac g2(s-2). $$ Thus Nguyen's lower bound $s\ge4$ 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 $g$.