Lagrangian Fibrations on Symplectic Varieties of Dimension 4

D. Markushevich and A. S. Tikhomirov introduced a 4-dimensional irreducible symplectic V-manifoldby constructing the relative compactified Prym variety of a family of curves with involution. We show thatthe Markushevich–Tikhomirov system specializes to a birational modification of a Beauville–Mukai systemmodulo a fibre-preserving involution induced by a two-torsion line bundle. This provides an explicit mapconsistent with a degeneration of the M–T system to a Nikulin variety as a Lagrangian fibration predictedin recent work of G. Nanni. We also utilize the weight formula for discriminant divisors of J. Ehrhard toevidence a similar relationship between the relative Prymian of E. Arbarello, G. Saccà, and A. Ferretti andthe quotient of a K3 [2] -type hyperkähler manifold by a group of order 4.