Birational geometry of Beauville-Mukai systems II: general theory in low ranks

Abstract

Via wall-crossing, we study the birational geometry of Beauville-Mukai systems on K3 surfaces with Picard rank one. We show that there is a class of walls which are always present in the movable cones of Beauville-Mukai systems. We give a complete description of the birational geometry of rank two Beauville-Mukai systems when the genus of the surface is small.

Birational geometry of Beauville-Mukai systems III: asymptotic behavior

“…When k = 1, Hilb 5 (S) is birational to the Beauville-Mukai system M (0, 2, −1). Its birational geometry has been studied in [Hel20] and [QS22b]. This proves (1).…”

“…We will call the subset of {(x, y) ∈ R 2 | y > 0} consisting of such pairs of (x, y) the xy-plane. We refer the reader to [BM14b,QS22b] for the wall and chamber structure on the xy-plane.…”

“…In our previous papers [QS22a,QS22b] we investigated the cases when N is of the form dn 2 for a positive integer n. In those cases Hilb N +1 (S) is birational (but not isomorphic) to a Beauville-Mukai system on S [Bea91, Muk84], a certain moduli space of pure dimension one sheaves on S, which admits the structure of a Lagrangian fibration. Moreover, the birational map can be decomposed into a sequence of stratified Mukai flops, where each flop yields a new birational model, a compact hyperkähler manifold birational to Hilb N +1 (S).…”