A general linear determinantal quartic in $\mathbb{ P}^4$ is nodal, non-$\mathbb{ Q}$-factorial and rational. We show that the family $\mathcal{ F}$ of such quartics also contains rational $\mathbb{ Q}$-factorial quartics, and that a generic member of $\mathcal{ F}$ can specialize to a rational non-$\mathbb{ Q}$-factorial double quadric. We describe the birational geometry of these three types of 3-folds, showing that it is governed by the extrinsic geometry of a curve $C\subset \mathbb{ P}^3$.
Categories: math.AG