Birational geometry of Beauville–Mukai systems III: Asymptotic behavior

Suppose that a Hilbert scheme of points on a K3 surface $S$ of Picard rank one admits a rational Lagrangian fibration. We show that if the degree of the surface is sufficiently large compared to the number of points, then the Hilbert scheme is the unique hyperkähler manifold in its birational class. In particular, the Hilbert scheme is a Lagrangian fibration itself, which we realize as coming from a (twisted) Beauville–Mukai system on a Fourier–Mukai partner of $S$. We also show that when the degree of the surface is small our method can be used to find all birational models of the Hilbert scheme.