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Let $(R,\mathfrak{m},k)$ be a discrete valuation ring with fraction field $K$. Let $X/K$ be a smooth projective variety.

Let $\mathcal{X},\mathcal{X}'$ be smooth projective models of $X$ over $R$. Let $X_0,X'_0$ be special fibers of $\mathcal{X},\mathcal{X}'$.

(1) Are there examples that the chow rings $\mathrm{CH}^*(X_0)$ and $\mathrm{CH}^*(X_0')$ are not isomorphic?

(2) If $\mathrm{CH}^*(X_0)$ is isomorphic to $\mathrm{CH}^*(X_0')$, can we always choose an isomorphism that is compatible with the specialization maps $\rho\colon\mathrm{CH}^*(\mathcal{X})\to\mathrm{CH}^*(X_0)$ and $\rho'\colon\mathrm{CH}^*(\mathcal{X})\to\mathrm{CH}^*(X_0')$?

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    $\begingroup$ I am replacing my previous comment with a precise reference. Your question is answered by Liedtke's analysis of the moduli stack of polarized Enriques surface over $\text{Spec}(\mathbb{Z})$, cf. arxiv.org/pdf/1007.0787.pdf Theorem 4.14 is particularly relevant since the $\mathbb{G}_m$-quotient of the scheme $\text{Spec} \mathbb{Z}[x,y]/\langle xy-2 \rangle \setminus \{\langle 2,\overline{x},\overline{y}\rangle\}$ is the line with doubled origin. $\endgroup$
    – Jason Starr
    Commented Jan 1, 2022 at 12:46
  • $\begingroup$ . . . I will try to expand this to an answer when I have time. (I am happy if somebody else writes an answer before that!) $\endgroup$
    – Jason Starr
    Commented Jan 1, 2022 at 12:49
  • $\begingroup$ My previous comment is wrong. The moduli space is indeed over the nonseparated “arithmetic line”. However it is very nonproper. Every valuation ring that extends over one of the doubled points in characteristic 2, in fact, does not extend over the other doubled point. $\endgroup$
    – Jason Starr
    Commented Jan 2, 2022 at 12:04
  • $\begingroup$ Thanks! Would it be possible to find such example for quardic surfaces? (If we can pick degenerations with separated/Galois conjugate rulings?) $\endgroup$
    – user39380
    Commented Jan 2, 2022 at 12:16
  • $\begingroup$ No, it is not possible to find such an example for a smooth quadric surface (or a "twist" of a smooth quadric surface). The cycle class map is injective in this case, and the smooth and proper base change theorems allow to describe the etale cohomology in terms of the generic fiber with its Galois action (so the same for both possible specializations). That is also another way to see that my previous comment is wrong. Any valid example would "arise" from the kernel of the cycle class map. $\endgroup$
    – Jason Starr
    Commented Jan 2, 2022 at 13:40

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