Double EPW sextics associated to Gushel-Mukai surfaces

Abstract: Works by O'Grady allow to associate to a 2-dimensional Gushel-Mukai variety, which is a K3 surface, a double EPW sextic. We characterize the K3 surfaces whose associated double EPW sextic is smooth. As a consequence, we are able to produce symplectic actions on some families of smooth double EPW sextics which are hyper-K\"ahler manifolds. We also provide bounds for the automorphism group of Gushel-Mukai varieties in dimension 2 and higher.