In this paper we generalise F. Catanese
s work on singular (
/2
)
2-covers to arbitrary finite abelian covers of algebraic surfaces. We determine the contribution of singularities to the invariants
,
K
2,
q,
p
g
,
P
n
and the canonical sheaf. We use these computations to construct a surface of general type with birational canonical map
1,
p
g
=4 and
K
2
=31.
Mathematics Subject Classification (2000):
14J29, 14J17, 14J25, 14E20