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We study wall-crossing for the Beauville–Mukai system of rank three on a general genus two K3 surface. We show that such a system is related to the Hilbert scheme of ten points on the surface by a sequence of flops, whose exceptional loci can be described as Brill–Noether loci. We also obtain Brill–Noether type results for sheaves in the Beauville–Mukai system.
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Birational geometry of Beauville–Mukai systems III: Asymptotic behavior
Abstract Suppose that a Hilbert scheme of points on a K3 surface 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 . 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.
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- PAR ID:
- 10544948
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- ISSN:
- 0024-6093
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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