  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.
