A Finiteness Theorem for Quartic K3-Fibred Calabi--Yau Threefolds in Scrolls

We study a restricted form of Gross's finiteness problem for algebraic minimal Calabi--Yau threefolds: those fibred by quartic K3 surfaces and realised as anticanonical hypersurfaces in stacky scrolls $\PP^1\times[\PP^3/\ZZ_n]$. Beyond the ten straight-scroll families of \cite[Table~1]{MboyaSzendroi2023}, we introduce orbifold scrolls $(\PP^1\times\PP^3)/\ZZ_n$ and apply the Reid--Shepherd-Barron--Tai criterion to determine which admit canonical anticanonical hypersurfaces. Only finitely many weight vectors arise for each $n$; we classify $n=1,2,3$ completely and compute all Hodge numbers, giving fourteen deformation families in total. Engel, Filipazzi, Greer, Mauri and Svaldi \cite{EngelFilipazziGreerMauriSvaldi2025} have since established boundedness for fibred Calabi--Yau threefolds in general, settling the existence question this restricted case exemplifies; what remains, and what we supply here, is the explicit classification: weight vectors, singularity types, and Hodge data.