Moduli spaces of model surfaces with one-dimensional complex tangent

We consider the moduli spaces $\mathcal{M}(n,K)$ that parametrize the set of mutually inequivalent model surfaces. We construct the spaces $\mathcal{M}(1,K)$ for $K\le13$ and prove some results on the structure of $\mathcal{M}(1,K)$ for arbitrary $K$.