Example of such an affine scheme

Question

I am stucking on construction an affine $\Bbb Z$-scheme $X$ such that $X_{\Bbb F_2}$ is irreducible but not reduced, $X_{\Bbb F_3}$ is reduced but not irreducible, and $X_\Bbb C$ is reduced and irreducible.

That is I need to find a proper number of variables and a relation set $T$ such that for $Spec(\Bbb Z[x_s:s\in S]/\langle T\rangle)$, such that if we replace the scaler $\Bbb Z$ by $\Bbb Z_2,\Bbb Z_3$ and $\Bbb C$, we have the affine scheme with desired property.

I cannot find a way that these conditions could be satisfied simutanously. Any ideas for that? Thanks in advance!

Answer 1

Something like $\operatorname{Spec}(\mathbb{Z}[x, y] / \langle 2x^3 + 3x^2 - 2x - 6y \rangle )$ should work.

Answer 2

Hint: Try just taking a disjoint union of schemes $A\coprod B\coprod C$, where $A$ is a scheme over $\mathbb{F}_2$, $B$ is a scheme over $\mathbb{F}_3$, and $C$ is a scheme over $\mathbb{Z}[1/6]$.

A specific example that works is hidden below.

Let $A=\operatorname{Spec}\mathbb{F}_2[x]/(x^2)$, $B=\operatorname{Spec}\mathbb{F}_3^2$, and $C=\operatorname{Spec}\mathbb{Z}[1/6]$, and let $X=A\coprod B\coprod C$. Explicitly, $X=\operatorname{Spec} (\mathbb{F}_2[x]/(x^2)\times \mathbb{F}_3^2\times\mathbb{Z}[1/6])$.