  Let F be a flat family of projective schemes, whose geometric generic fiber
is reduced and irreducible. We give conditions on a special fiber (a "limit" of
the family) to guarantee that it too is reduced. These conditions often imply
also that the generic fiber is normal. The conditions are particularly easy to
check in the setup of a "geometric vertex decomposition" [Knutson-Miller-Yong
'07].
  The primary tool used is the corresponding limit _branchvariety_
[Alexeev-Knutson '06], which is reduced by construction, and maps to the limit
subscheme; our technique is to use normality to show that the branchvariety map
must be an isomorphism.
  As a demonstration, we give an essentially naive proof that Schubert
varieties in finite type are normal and Cohen-Macaulay. The proof does not
involve any resolution of singularities or cohomology-vanishing techniques
(e.g. appeal to characteristic p).
