If a degenerating threefold has canonical singularities, the moduli space of pairs of P^3 and hypersurfaces is smooth at the corresponding point.
Koll´ ar,Singularities of pairs, In: Algebraic geometry—Santa Cruz 1995, 221–287, Proc
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Deformation of pairs of $\mathbb{P}^3$ and hypersurfaces
If a degenerating threefold has canonical singularities, the moduli space of pairs of P^3 and hypersurfaces is smooth at the corresponding point.