  What kind of reduced monomial schemes can be obtained as a Gr\"obner
degeneration of a smooth projective variety? Our conjectured answer is: only
Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial
complexes. This would imply, in particular, that only curves of genus zero have
such a degeneration. We prove this conjecture for degrevlex orders, for
elliptic curves over real number fields, for boundaries of cross-polytopes, and
for leafless graphs. We discuss consequences for rational and F-rational
singularities of algebras with straightening laws.
