Surfaces on the Severi line in positive characteristics

Abstract

Let $X$ be a minimal surface of general type over an algebraically closed field $\mathbf{k}$ of $\mathrm{char}.(\mathbf{k})=p\ge 0$. If the Albanese morphism $a_X:X\to {\mathrm{Alb}}_X$ is generically finite onto its image, we formulate a constant $c(X,L)\ge 0$ for a very ample line bundle $L$ on ${\mathrm{Alb}}_X$ such that $c(X,L)=0$ if and only if $\dim {\mathrm{Alb}}_X=2$ and $a_X:X\to {\mathrm{Alb}}_X$ is a double cover. A refined Severi inequality $$K_X^2\ge (4+\min \{c(X,L),\frac{1}{3}\})\chi ({\mathcal{O}}_X)$$ is proved. Then we prove that $K_X^2=4\chi ({\mathcal{O}}_X)$ if and only if the canonical model of $X$ is a flat double cover of an Abelian surface.

some typos corrected