发表机构
Shanghai Normal University（上海师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在正特征下证明半稳定纤维化曲面的Arakelov不等式，并构造达到Nguyen下界和Hodge-Witt最优性的亏格2例子，进而导出典范类与Szpiro型不等式。
AI 中文摘要
设$f:S\to C$为特征$p>0$的代数闭域上亏格$g\ge2$的、具有光滑且几何连通的通用纤维的相对极小半稳定纤维化。令$\Sigma\subset C$表示纤维为奇异的点集，$s=|\Sigma|$。若总曲面$S$是Hodge Witt的且$s\ge2$，我们证明经典Arakelov不等式$$ \text{deg}\\, f_*\omega_{S/C}\le \frac g2 \text{deg}\\, \Omega_C^1(\log\Sigma), $$其中$\omega_{S/C}$表示相对对偶层。对于$C=\mathbb P^1$，我们得到更强的估计$$ \text{deg}\\, f_*\omega_{S/\mathbb P^1}\le \frac g2(s-2)-\frac12 b_1(S), $$其中$b_1(S)$是$S$的第一Betti数。我们还构造了特征$5$中$\mathbb P^1$上具有恰好$4$个奇异纤维的亏格$2$半稳定纤维化。其总曲面是Hodge Witt的，且$$ \text{deg}\\, f_*\omega_{S/\mathbb P^1}=2=\frac g2(s-2)。 $$因此Nguyen的下界$s\ge4$和Hodge--Witt Arakelov不等式都是最优的。作为应用，我们推导出典范类不等式和系数关于$g$线性的Szpiro型不等式。
英文摘要
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 $g$.
Comments28 pages, comments are welcome!
