Automatically Reduced Degenerations of Automatically Normal Varieties

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 naı̈ve 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).