The Stacks project

Remark 82.2.3. In Situation 82.2.1 if $X/B$ is good, then $X$ is integral (Spaces over Fields, Definition 72.4.1) if and only if $X$ is reduced and $|X|$ is irreducible. Moreover, for any point $\xi \in |X|$ there is a unique integral closed subspace $Z \subset X$ such that $\xi $ is the generic point of the closed subset $|Z| \subset |X|$, see Spaces over Fields, Lemma 72.4.7.