Smoothing of multiple structures on embedded Enriques manifolds

Abstract

We show that given an embedding of an Enriques manifold of index $d$ in a large enough projective space, there will exist embedded multiple structures with conormal bundle isomorphic to the trace zero module of the universal covering map, the universal cover being either a hyperk\"ahler or a Calabi-Yau manifold. We then show that these multiple structures (also known as $d$-ropes) can be smoothed to smooth hyperk\"ahler or Calabi-Yau manifolds respectively. Hence we obtain a flat family of hyperk\"ahler (or Calabi-Yau) manifolds embedded in the same projective space which degenerates to an embedded $d$-rope structure on the given Enriques manifold of index $d$. The above shows that these $d$-rope structures on the embedded Enriques manifold are points of the Hilbert scheme containing the fibres of the above family. We show that they are smooth points of the Hilbert scheme when $d=2$.