The Canonical Map and Horikawa Surfaces in Positive Characteristic

TL;DR

This paper extends inequalities and classifies Horikawa surfaces in positive characteristic, analyzing their moduli, cohomology, and lifting properties, revealing new geometric and topological insights.

Contribution

It provides a classification of Horikawa surfaces in positive characteristic, describes their moduli spaces, and investigates their cohomological and lifting properties.

Findings

Classification of surfaces on the Noether lines in positive characteristic

Description of moduli spaces and inseparable canonical maps

Computation of Betti, de Rham, and crystalline cohomology

Abstract

We extend fundamental inequalities related to the canonical map of surfaces of general type to positive characteristic. Next, we classify surfaces on the Noether lines, i.e., even and odd Horikawa surfaces, in positive characteristic. We describe their moduli spaces and the subspaces formed by surfaces whose canonical maps are inseparable. Moreover, we compute their Betti-, deRham- and crystalline cohomology. Finally, we prove lifting to characteristic zero and show that the moduli spaces are topologically flat over the integers.