  We construct degenerations of Mukai varieties and linear sections thereof to
special unobstructed Fano Stanley-Reisner schemes corresponding to convex
deltahedra. This can be used to find toric degenerations of rank one index one
Fano threefolds. In the second part we find many higher dimensional
unobstructed Fano and Calabi-Yau Stanley-Reisner schemes. The main result is
that the Stanley-Reisner ring of the boundary complex of the dual polytope of
the associahedron has trivial $T^2$.
