  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.
