  We study N=(4,4) superconformal field theories with left and right central
charge c=6 which allow geometric interpretations on specific quartic
hypersurfaces in CP^3. Namely, we recall the proof that the Gepner model (2)^4
admits a geometric interpretation on the Fermat quartic and give an independent
cross-check of this result, providing a link to the "mirror moonshine
phenomenon" on K3. We clarify the role of Shioda-Inose structures in our proof
and thereby generalize it: We introduce "very attractive quartics" and show how
on each of them a superconformal field theory can be constructed explicitly.
