  A fake quadric is a smooth minimal surface of general type with the same
invariants as the quadric in P^3, i.e. K^2=2c_2=8 and q=p_g=0. We study here
quaternionic fake quadrics i.e. fake quadrics constructed arithmetically by
using some quaternion algebras over real number fields.
  We provide examples of quaternionic fake quadrics X with a non-trivial
automorphism group and compute the invariants of the minimal desingularisation
of the quotient of X by this group. In that way we obtain minimal surfaces of
general type Z with q=p_g=0 and K^2=4,2 which contain the maximal number of
disjoint nodal curves. We then prove that if a surface of general type has the
same invariant as Z and same number of nodal curves, we can construct
geometrically a surface of general type with K^2=2c_2=8.

Algebraic Geometry