Question

(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?

Answer 1

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 you can find some sufficient conditions. For instance, the lifting property for finite generically étale morphisms of semi-stable curves are characterized.

Answer 2

I don't think you can extend an arbitrary morphism, even if the curves are prestable.

For instance, if $C_0$ is a quartic with one node and the generic element is a smooth quartic then you do have a morphism from a smooth curve of genus $2$ to $C_0$ which does not extend, since any extension would map a curve of genus $2$ to a curve of genus $3$.