Let $X$ be a projective scheme over a finite field $k$ (say, $\mathbb F_p$), I would like to lift it to a projective scheme $X'$ over char $0$ ring $R$ (say $\mathbb Z$) with a surjective map $R \to k$ such that $X'_k \cong X$. I think the following should work:
There will be some $\mathbb P^n_k$ for which $X$ is defined as a closed subscheme by a homogeneous ideal $I \subset k[x_0,\dots,x_n]$. We can arbitrarily lift the generators of $I$ to polynomials in $R[x_0,\dots,x_n]$ and hence define a homogenous ideal $I'$ over $R$. Then, define $X'$ to be the closed subscheme defined by $I'$.
I would now like to show that the specialization of $X'$ to $k$ would recover $X$ but I don't think this is true because of the counterexample here: https://mathoverflow.net/questions/25337/lifting-varieties-to-characteristic-zero.
I think the problem is that we would end up getting a little bit more than $X$ but I am having a hard time figuring out why or finding an example where this happens. Any help?