Arakelov inequalities for fibered surfaces in positive characteristic

Abstract

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 $\Sigma \subset C$ denote the set of points over which the fibers are singular, with $s=|\Sigma |$. If the total surface $S$ is Hodge Witt and $s\ge 2$, we prove the classical Arakelov inequality $$\text{deg}{f}_{\ast }{\omega }_{S/C}\le \frac{g}{2}\text{deg}{\Omega }_C^1(\log \Sigma ),$$ where ${\omega }_{S/C}$ denotes the relative dualizing sheaf. For $C={\mathbb{P}}^1$, we obtain the stronger estimate $$\text{deg}{f}_{\ast }{\omega }_{S/{\mathbb{P}}^1}\le \frac{g}{2}(s-2)-\frac{1}{2}{b}_1(S),$$ 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 $$\text{deg}{f}_{\ast }{\omega }_{S/{\mathbb{P}}^1}=2=\frac{g}{2}(s-2).$$ 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 $g$.

arXiv comment28 pages, comments are welcome!