On cubic elliptic varieties

TL;DR

This paper investigates the relationship between the finiteness of the Mordell-Weil group and the Cox ring of certain elliptic fibrations derived from cubic hypersurfaces, providing criteria and presentations for the Cox ring.

Contribution

It establishes an equivalence between the finiteness of the Mordell-Weil group and the finite generation of the Cox ring for these elliptic varieties, and offers explicit Cox ring presentations.

Findings

Mordell-Weil group is finite iff Cox ring is finitely generated.

Provides explicit Cox ring presentations when finitely generated.

Characterizes elliptic fibrations from cubic hypersurfaces.

Abstract

Let X->P^(n-1) be an elliptic fibration obtained by resolving the indeterminacy of the projection of a cubic hypersurface Y of P^(n+1) from a line L not contained in Y. We prove that the Mordell-Weil group of the elliptic fibration is finite if and only if the Cox ring of X is finitely generated. We also provide a presentation of the Cox ring of X when it is finitely generated.