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

TL;DR

This paper classifies degree 1 elliptic threefolds with constant j-invariant, detailing their Mordell-Weil groups and describing certain singular hypersurfaces with no variation of Hodge structure.

Contribution

It provides a complete classification of such elliptic threefolds and their Mordell-Weil groups, especially when the j-invariant is nonzero.

Findings

Classification of Mordell-Weil groups for these threefolds

Explicit description of elliptic threefolds with constant j-invariant

Identification of singular hypersurfaces with no variation of Hodge structure

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 $\Ps(2,3,1,1,1)$ that admit no variation of Hodge structure.