Chern slopes of surfaces of general type in positive characteristic

TL;DR

This paper constructs minimal surfaces of general type over algebraically closed fields of positive characteristic with prescribed Chern slope ratios, demonstrating their density in the interval and exploring properties like fundamental group and Picard scheme.

Contribution

It proves the density of Chern slopes in the interval [2, ∞) for surfaces over fields of positive characteristic, including cases with trivial Picard scheme.

Findings

Chern slopes are dense in [2, ∞) for positive characteristic surfaces.

Existence of surfaces with prescribed fundamental group and Chern slope ratio.

Surfaces with trivial Picard scheme also densely populate the interval.

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic $p>0$, and let $C$ be a nonsingular projective curve over $\mathbb{K}$. We prove that for any real number $x \geq 2$, there are minimal surfaces of general type $X$ over $\mathbb{K}$ such that a) $c_1^2(X)>0, c_2(X)>0$, b) $\pi_1^{\text{\'et}}(X) \simeq \pi_1^{\text{\'et}}(C)$, c) and $c_1^2(X)/c_2(X)$ is arbitrarily close to $x$. In particular, we show 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 $H^1(X,\mathcal{O}_X)=0$, this 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 $p$.