Explicit birational geometry of determinantal quartic 3-folds

Abstract

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 prove that the birational geometry of these three types of 3-folds is governed by the extrinsic geometry of a curve $C\subset \mathbb{P}^3$ of degree 10 and genus 11.