Toric Calabi-Yau hypersurfaces fibered by weighted K3 hypersurfaces

In response to a question of Reid, we find all anti-canonical Calabi--Yauhypersurfaces $X$ in toric weighted $\PP^3$-bundles over $\PP^1$ wherethe general fiber of $X$ over $\PP^1$ is a weighted K3 hypersurface.This gives a direct generalization of Reid's discovery of the 95families of weighted K3 hypersurfaces in~\cite{mR80}. We also treat thecase where $X$ is fibered over $\PP^2$ with general fiber agenus one curve in a weighted projective~plane.

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Published 1 January 2009