Lemma 33.6.8. Let k be a field. Let X be a scheme over k.
If x' \leadsto x is a specialization and X is geometrically reduced at x, then X is geometrically reduced at x'.
If x \in X such that (a) \mathcal{O}_{X, x} is reduced, and (b) for each specialization x' \leadsto x where x' is a generic point of an irreducible component of X the scheme X is geometrically reduced at x', then X is geometrically reduced at x.
If X is reduced and geometrically reduced at all generic points of irreducible components of X, then X is geometrically reduced.
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