Marked and Labelled Gushel–Mukai Fourfolds

We prove that the moduli stacks of marked and labelled Hodgespecial Gushel–Mukai fourfolds are isomorphic. As an application, we construct rational maps from the stack of Hodge-special Gushel–Mukai fourfolds of discriminant d to the moduli space of (twisted) degree-d polarized K3 surfaces. We use these results to prove a counting formula for the number of fourdimensional fibers of Fourier–Mukai partners of very general Hodge-special Gushel–Mukai fourfolds with associated K3 surface, and a lower bound for this number in the case of a twisted associated K3 surface.