On the classification of degree 1 elliptic threefolds with constant j-invariant

Abstract

We describe the possible Mordell–Weil groups for degree 1 elliptic threefold with rational base and constant $j$-invariant. Moreover, we classify all such elliptic threefolds if the $j$-invariant is nonzero. We can use this classification to describe a class of singular hypersurfaces in P(2, 3, 1, 1, 1) that admit no variation of Hodge structure (Remark 9.3).