  We analyse the diagonal quotient for products of certain Artin--Schreier
curves. The smooth models are almost always surfaces of general type, with
Chern slopes tending asymptotically to 1. The calculation of numerical
invariants relies on a close examination of the relevant quotient singularity
in characteristic p. It turns out that the canonical model has q-1 rational
double points of type A_{q-1}, and embeds as a divisor of degree q in P^3,
which is in some sense reminiscent of the classical Kummer quartic.
