On the geometry of singular EPW cubes

EPW cubes form a locally complete family of smooth projective hyper-Kähler varieties of dimension 6, constructed by Iliev–Kapustka–Kapustka–Ranestad.\ Their construction and behavior share a lot of similarities with the double EPW sextics constructed by O’Grady.\ Adapting the methods of O’Grady, we construct a projective smooth small resolution of singular EPW cubes.