A remarkable class of elliptic surfaces of amplitude 1 in weighted projective space

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. We added an application to certain compactifications of moduli spaces of surfaces of general type with $K^2=1$, $p_g=2$ and $q=0$, as well as detailed SageMath-calculations. The appendix written by Wim Nijgh shows that the general member of the type 1 and type 2 elliptic family has "trivial" Picard lattice, i.e. is spanned by fiber components and a multisection.