(I'm happy to work over an algebraically closed field....)


Let $\mathcal{C} \rightarrow Spec (R)$ be a (flat) family of proper, prestable curves where $R$ is a DVR. Suppose the generic fiber is smooth and the special fiber, $C\_0$, is reduced but may be reducible. 


Given a finite map of curves $f\_0: D\_0 \rightarrow C\_0$ with $D\_0$ also prestable, can this be extended to some map on some family?


That is, is there a flat family of proper curves $\mathcal{D} \rightarrow Spec(R)$ and an $R$-morphism $f: \mathcal{D} \rightarrow \mathcal{C}$ which reduces to $f\_0$ on the special fiber?


Perhaps such an extension is possible only after a ramified cover of $Spec(R)$?


If so, can it be arranged that the generic fiber of $\mathcal{D}$ is smooth?



Extending maps of curves
Stated as such, the question is really too vague. 
For example, if $C\_0, D\_0$ are smooth of genus bigger than 1, and $f\_0$ is purely inseparable, then $f\_0$ can not lift to $R$ (even after extention) if $R$ has characteristic zero. Actually, as the generic fibers of $\mathcal C$ and $\mathcal D$ are equal to $g(C\_0)=g(D\_0)$, this will contradict Riemann-Hurwitz formula. 


In [this survey paper](http://www.math.u-bordeaux1.fr/~liu/articles/tokyo.ps) you can find some sufficient conditions. For instance, the lifting property for finite generically étale morphisms of semi-stable curves are characterized. 


