Arakelov inequalities for fibered surfaces in positive characteristic

Source 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 $Σ\subset C$ denote the set of points over which the fibers are singular, with $s=|Σ|$. If the total surface $S$ is Hodge Witt and $s\ge2$, we prove the classical Arakelov inequality $$\text{deg}\, f_*ω_{S/C}\le \frac g2 \text{deg}\, Ω_C^1(\logΣ),$$ where $ω_{S/C}$ denotes the relative dualizing sheaf. For $C=\mathbb P^1$, we obtain the stronger estimate $$\text{deg}\, f_*ω_{S/\mathbb P^1}\le \frac g2(s-2)-\frac12 b_1(S),$$ 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 $$\text{deg}\, f_*ω_{S/\mathbb P^1}=2=\frac g2(s-2).$$ 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 $g$.