Let $X$ be a projective scheme over $Spec(\mathbb{Z})$. Let $X_{p}$ be the reduction at $p$ of $X$. If for any prime $p$, $X_{p}$ is normal, can we deduce $X$ is normal? Or any counterexamples?
$\begingroup$
$\endgroup$
4
-
$\begingroup$ Do you mean "if for every $p$, $X_p$ is normal" or "If for at least one $p$, $X_p$ is normal"? $\endgroup$– Will SawinCommented Sep 27, 2022 at 17:05
-
$\begingroup$ @Will Sawin I means for every p. Maybe I need add a condition such that $X$ is flat over $\mathbb{Z}$ $\endgroup$– fool rabbitCommented Sep 27, 2022 at 17:53
-
$\begingroup$ What is the ring where the prime ideal $p$ "lives"? $\endgroup$– Armando j18eosCommented Sep 27, 2022 at 17:59
-
4$\begingroup$ Yes, flatness is needed, or else you can take the union of $\mathbb P^1_{\mathbb F_p}$ and $\operatorname{Spec} \mathbb Z$, glued at an $\mathbb F_p$-point of $\mathbb P^1_{\mathbb F_p}$, whose mod $p$ fiber is $\mathbb P^1$, normal, and whose mod $q$ fiber is a point for any other point $q$, but is not normal. $\endgroup$– Will SawinCommented Sep 27, 2022 at 19:07
Add a comment
|