Birational geometry of Beauville-Mukai systems III: asymptotic behavior

TL;DR

This paper investigates the birational geometry of Hilbert schemes of points on K3 surfaces, establishing conditions under which these schemes are unique hyperkähler manifolds with Lagrangian fibrations, and describing their birational models.

Contribution

It demonstrates that for large surface degrees, the Hilbert scheme is uniquely determined within its birational class and arises from a Beauville-Mukai system, extending understanding of their geometric structure.

Findings

Hilbert schemes are unique hyperkähler manifolds with Lagrangian fibrations for large degrees.

The Lagrangian fibration can be realized via a (twisted) Beauville-Mukai system.

Method to find all birational models when the surface degree is small.

Abstract

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 hyperk\"ahler 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.