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.
Marked and labelled Gushel-Mukai fourfolds / E. Brakkee, L. Pertusi (PROGRESS IN MATHEMATICS). - In: Rationality of Varieties / [a cura di] G. Farkas, G. van der Geer, M. Shen, L. Taelman. - [s.l] : Springer, 2021. - ISBN 978-3-030-75421-1. - pp. 129-146
Marked and labelled Gushel-Mukai fourfolds
L. Pertusi
2021
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.
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