Q:

Lifting of families of curves to characteristic 0

Let $k$ be a finite field, $X_0$ be a smooth affine variety over $k$ and $C\rightarrow X$ a smooth projective family of curves of genus $\geq 2$.
By a result of Elkik we can always lift $X_0$ to a smooth affine scheme over $W(k)$. 
It seems that, up to replace $X_0$ with an open subset, one can always lift $C_0\rightarrow X_0$ to a smooth projective family of curves $C\rightarrow X$.
Does anyone have a reference for this fact?

A:

I am just posting my comment as an answer.  This type of lifting is "classical".  In the article linked above with Chenyang Xu, we needed bounds on the lift, to deduce height bounds for some rational points.  So we worked out in detail the classical result.
The result that I called "Chow's Lemma for Stacks" is not actually Chow's Lemma for Stacks (sorry).  Rather, it is Théorème 6.3, p. 49 of the following.
MR1771927 
Laumon, Gérard; Moret-Bailly, Laurent 
Champs algébriques. 
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Volume 39. 
Springer-Verlag, Berlin, 2000. xii+208 pp. 
ISBN: 3-540-65761-4 
For each field $k$ of finite characteristic $p$, denote by $W(k)$ a complete, local, Noetherian DVR with uniformizing element $p$ such that $W(k)/pW(k)$ equals $k$.  Let $\mathcal{M}$ be a smooth stack over $\text{Spec}\ W(k)$.  Let $X_0$ be a connected, smooth, quasi-projective $k$-scheme.  By hypothesis, there exists a locally closed immersion of $k$-schemes, $$i:X_0 \to \mathbb{P}^n_k.$$  Let $\zeta:X_0\to \mathcal{M}$ be a $1$-morphism over $\text{Spec}\ W(k)$.  By Théorème 6.3, up to replacing $X_0$ by a dense open subscheme, there exists a factorization of $\zeta$, $$X_0\xrightarrow{z} M \xrightarrow{\phi} \mathcal{M},$$ where $M$ is a smooth, quasi-projective $W(k)$-scheme, where $\phi$ is a smooth $1$-morphism over $W(k)$, and where $z$ is a $W(k)$-morphism.
So now we are reduced to lifting over $W(k)$ the locally closed subscheme $X_0$ in the closed fiber of the smooth, quasi-projective $W(k)$-scheme $$\mathbb{P}^n_{W(k)}\times_{\text{Spec}\ W(k)} M.$$  This is "classical".  Since $X_0$ is smooth, the locally closed immersion is a regular embedding.  Thus, up to shrinking $X_0$ further, this closed subscheme is an open subscheme of a complete intersection of degree $d$ ample hypersurfaces in $\mathbb{P}^n_{W(k)}\times_{\text{Spec}\ W(k)} M$ for all $d\geq d_0$. (You can spend a lot of time trying to optimize degrees in this argument, but the OP did not ask for an "optimal lifting", just some lifting.)  The defining equations of those hypersurfaces have coefficients that are elements of $k$.  By lifting those coefficients to $W(k)$, we can lift the hypersurfaces over $W(k)$.  This lifts the complete intersection over $W(k)$.
As the commenters note, this clashes with intuition that lifts should preserve intersection numbers and other notions that definitely are preserved for lifts of projective schemes.  However, this is a well-known "pathology" of lifts of affine schemes.  We can find projective models of these lifts, but then the closed fiber will be reducible.  Those properties suggested by our intuition typically do occur for the closed fiber of a projective model.  However, the property may hold for some irreducible component of the closed fiber different from the component containing $X_0$. 

