Chern slopes of surfaces of general type in positive characteristic

Abstract

Let $\mathbf{k}$ be an algebraically closed field of characteristic $p>0$, and let $C$ be a nonsingular projective curve over $\mathbf{k}$. We prove that for any real number $x\ge 2$, there are minimal surfaces of general type $X$ over $\mathbf{k}$ such that (a) $c_1^2\left(X\right)>0$, $c_2\left(X\right)>0$, (b) ${\pi }_1^{\acute{e}t}\left(X\right)\simeq {\pi }_1^{\acute{e}t}\left(C\right)$, and (c) $c_1^2\left(X\right)/c_2\left(X\right)$ is arbitrarily close to $x$. In particular, we show the density of Chern slopes in the pathological Bogomolov–Miyaoka–Yau interval $(3,\infty )$ for any given $p$. Moreover, we prove that for $C={\mathbb{P}}^1$ there exist surfaces $X$ as above with $H^1(X,{\mathcal{O}}_X)=0$, that is, with Picard scheme equal to a reduced point. In this way, we show that even surfaces with reduced Picard scheme are densely persistent in $[2,\infty )$ for any given $p$.