Hypersurfaces in Mori dream spaces

TL;DR

This paper characterizes when the Cox ring of a hypersurface in a Mori dream space can be described as a quotient of the Cox ring of the ambient space, with applications to Calabi-Yau hypersurfaces and hypersurfaces containing linear subspaces.

Contribution

It provides necessary and sufficient conditions for Cox rings of hypersurfaces in Mori dream spaces to be quotients, extending understanding of their algebraic structure.

Findings

Cox ring of hypersurfaces can be explicitly described under certain conditions.

Applications to Calabi-Yau hypersurfaces in toric Fano fourfolds.

Results on hypersurfaces in projective space containing linear subspaces.

Abstract

Let X be a hypersurface of a Mori dream space Z. We provide necessary and sufficient conditions for the Cox ring R(X) of X to be isomorphic to R(Z)/(f), where R(Z) is the Cox ring of Z and f is a defining section for X. We apply our results to Calabi-Yau hypersurfaces of toric Fano fourfolds. Our second application is to general degree d hypersurfaces in P^n containing a linear subspace of dimension n-2.