Degenerations to unobstructed Fano Stanley-Reisner schemes (Q463965)

Mukai represented certain Fano threefolds as complete intersections in explicitely given ambient spaces (nowadays called Mukai varieties). In the present paper, the authors construct monomial degenerations of the latter and linear sections thereof. The limits are Stanley-Reisner schemes arising from simplicial complexes being the boundary of convex deltahedra. These are exactly those 3-dimensional polytopes having regular triangles as faces. Afterwards, the view point is changed. One starts with exactly these Stanley-Reisner schemes and understands the original Mukai varieties as generic deformations of them. The authors prove that this deformation theory is unobstructed, hence the Mukai varieties degenerate to every other variety specializing to these Stanley-Reisner schemes. Among them one finds toric ones provided by representing the simplicial complexes as unimodular triangulations of lattice polytopes. Finally, the authors apply this method to linear sections of \(\mathrm{Gr}(2,n)\), too.