Marked and labelled Gushel-Mukai fourfolds

Abstract: We prove that the moduli stacks of marked and labelled Hodge-special 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 4-dimensional 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.