A tower of complete moduli spaces of Calabi-Yau $n$-folds

We construct a sequence of complete moduli spaces $$E_0\subset E_1\subset E_2\subset \dots E_n\subset \dots ,$$ each of which is isomorphic to a weighted projective space. These spaces parameterize certain $n$-dimensional Calabi-Yau varieties associated with the Sylvester sequence $2,3,7,43,\dots$. They generalize the moduli space of elliptic curves ${\bar{M}}_{1,1}=\mathbb{P}(4,6)$ and Brieskorn’s family over ${\bar{F}}_{U\oplus E_8}^{\mathrm{B}\mathrm{B}}=\mathbb{P}(4,10,\dots ,42)$, the Baily-Borel compactification of the moduli space of $U\oplus E_8$-polarized K3 surfaces. We also study fibrations in such Calabi-Yau varieties, extending to higher dimensions the theory of elliptic surfaces.