In 1985, Beauville and Donagi showed by an explicit geometric construction that the variety of lines contained in a Pfaffian cubic hypersurface in ${\bf P}^5$ is isomorphic to a canonical desingularization of the symmetric self-product of a K3 surface (called its Hilbert square). Both of these projective fourfolds are hyperkähler (or symplectic): they carry a symplectic 2-form.
In 1998, Hassett showed by a deformation argument that this phenomenon occurs for countably many families of cubic hypersurfaces in ${\bf P}^5$.
Using the Verbitsky-Markman Torelli theorem and results of Bayer-Macri, we show these unexpected isomorphisms (or automorphisms) occur for many other families of hyperkähler fourfolds. This involves playing around with Pell-type diophantine equations. This is joint work with Emanuele MacrÃ.