  Surfaces of amplitude 1 in ordinary projective space are of general type, but
this need not be the case in weighted projective spaces. Indeed, there are 4
classes of quasi-smooth weighted hypersurfaces in $\mathbf{P}(1,2,a,b)$ of
amplitude 1 with an elliptic pencil cut out by hyperplanes. Their moduli spaces
are constructed, the monodromy of their universal families is determined as
well as their period maps. These all turn out to be non-injective. We analyse
the reason behind this, which for each type is different. For the two classes
that give properly elliptic surfaces this leads to a mixed Torelli-type theorem
as in the case of the Catanese-Kunev-Todorov surfaces.
