Godeaux, Campedelli, and surfaces of general type with χ=4 and 2≤K2≤8

We construct simply connected surfaces of general type with invariants χ(O)=4 and 2≤K2≤8. We use Q‐Gorenstein deformations in conjunction with explicit constructions that express the canonical rings by generators and relations. The canonical rings of the surfaces are described as projections. The whole construction is simplified by the use of key varieties based on Steiner 3‐folds. As a consequence of the construction we find two families, each family in a different connected component of the moduli stack M¯2,1, and each linking a Campedelli surface with a Godeaux surface.