On Cox Rings of Calabi-Yau hypersurfaces

TL;DR

This paper investigates Cox rings of smooth Calabi-Yau hypersurfaces in toric Fano varieties, identifying conditions for finite generation and exploring their automorphism groups and cone conjectures.

Contribution

It provides explicit Cox ring presentations, characterizes when hypersurfaces are Mori dream spaces, and examines the automorphism groups and cone conjectures in specific cases.

Findings

Identified configurations where hypersurfaces are Mori dream spaces.

Established conditions for infinite birational automorphism groups.

Proved the Morrison-Kawamata cone conjecture for certain non-Mori dream examples.

Abstract

We study the Cox rings of smooth anticanonical Calabi-Yau hypersurfaces in smooth toric Fano varieties. Using the combinatorics of primitive pairs of the ambient Fano polytope and the description of Cox rings of embedded varieties via localizations, we identify several configurations for which the hypersurface is a Mori dream space and obtain explicit presentations of its Cox ring. We also exhibit combinatorial configurations forcing the birational automorphism group to be infinite, yielding in dimensions three and four a dichotomy between finite generation of the Cox ring and infinite birational automorphism group. Finally, for a class of non-Mori dream examples, we prove the Morrison-Kawamata cone conjecture for the movable cone.