On cubic elliptic varieties

Abstract

Let π : X → P n − 1 \pi \colon X\to \mathbb {P}^{n-1} be an elliptic fibration obtained by resolving the indeterminacy of the projection of a cubic hypersurface Y Y of P n + 1 \mathbb {P}^{n+1} from a line L L not contained in Y Y . We prove that the Mordell-Weil group of π \pi is finite if and only if the Cox ring of X X is finitely generated. We also provide a presentation of the Cox ring of X X when it is finitely generated.