Birational geometry of the intermediate Jacobian fibration of a cubic fourfold

We show that the intermediate Jacobian fibration associated to any smooth cubic fourfold $X$ admits a hyper-K\"ahler compactification $J(X)$ with a regular Lagrangian fibration $J \to \mathbb P^5$. This builds upon arXiv:1602.05534, where the result is proved for general $X$, as well as on the degeneration techniques on arXiv:1704.02731 and techniques from the minimal model program. We then study some aspects of the birational geometry of $J(X)$: for very general $X$ we compute the movable and nef cones of $J(X)$, showing that $J(X)$ is not birational to the twisted version of the intermediate Jacobian fibration arXiv:1611.06679, nor to an OG$10$-type moduli space of objects in the Kuznetsov component of $X$; for any smooth $X$ we show, using normal functions, that the Mordell-Weil group $MW(\pi)$ of the abelian fibration $\pi: J \to \mathbb P^5$ is isomorphic to the integral degree $4$ primitive algebraic cohomology of $X$, i.e., $MW(\pi) = H^{2,2}(X, \mathbb Z)_0$.