The Canonical Map and Horikawa Surfaces in Positive Characteristic

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, that is, 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.