A generalization of elliptic curves to higher dimensions

Of course, there are many generalizations of elliptic curves. The one we consider here is a certain class of n-dimensional Calabi-Yau hypersurfaces in a weighted projective space, naturally associated with the Sylvester sequence $2,3,7,43,...,s_n$. The moduli space of such hypersurfaces is a weighted projective space itself. The case of $n=1$ for the Sylvester numbers 2,3 is the familiar case of elliptic curves in the Weierstrass form, and its compactified moduli space is the weighted projective line $P(4,6)$. 

 

For any n, we prove that the moduli space of pairs $(X,D)$ of such Calabi-Yau hypersurfaces $X$ augmented with a hyperplane $D$ at infinity is a connected component of the KSBA moduli space of stable pairs. A side result is a generalization of the theory of elliptic surfaces to higher dimensions. Based on https://arxiv.org/abs/2511.16562.