# Extending rational maps of nodal curves

Let $$R$$ be a discrete valuation ring with fraction field $$K$$ and $$C, D$$ two nodal (=prestable) curves over $$\operatorname{Spec} R$$. If I have a map $$C_K \to D_K$$ between the restriction of the curves to the generic point $$\operatorname{Spec} K$$,

1. can I blow up $$C$$ to extend this to a map $$C \to D$$ over $$\operatorname{Spec} R$$?

2. If I blow up $$C$$ to do this, can I ensure that the result remains a nodal curve and the generic fiber $$C_K$$ is unchanged?

If it helps, the map $$C_K \to D_K$$ is a "partial stabilization" -- it contracts $$\mathbb{P}^1$$'s.

Remark: This is pretty clear for smooth curves, and well documented in the stacks project.

(My plan of attack so far:

A) Stacks 0BX7 should let me extend the map away from a finite set of closed points in the special fiber of $$C$$.

B) I believe these closed points should be nodes that get smoothed out in the generic fiber -- if the node persists, then maybe I can define the map at the node via specialization of the two generic points without a problem.

C) If the node gets smoothed out, this is etale-locally pulled back from $$\mathbb{A}^2 \to \mathbb{A}^1$$ and one can see directly the blowup at this node remains nodal.

D) By taking the closure $$\Gamma$$ of the graph of the rational map $$C \dashrightarrow D$$, I can argue that the projection $$\Gamma \to C$$ is a blowup at nodes in the special fiber that get smoothed out, since the map can be extended already away from those points.

I don't completely believe any of these steps -- this is more of a sketch than a proof. I'd very much like citations and references to any argument, esp. to the stacks project. )

• I should mention that I'm over the complex numbers and the map $C \to D$, if it exists, would also be a partial stabilization. Jul 22, 2020 at 19:55

After lots of help from others, I realized the question boils down to some sort of stable reduction, which is a properness statement for the moduli of stable or relative stable maps $$\overline{M}(X/V)$$.
The isomorphism $$C_K \simeq D_K$$ provides a point $$C_K \to C_K \times D_K$$ over $$K$$ of $$\overline{M}(C \times D/\operatorname{Spec} R)$$. Properness (=stable reduction) extends this to a map $$C' \to C \times D$$ over $$\operatorname{Spec}R$$. The maps $$C' \to C, C' \to D$$ must be partial stabilizations because that's a closed condition in the base, and they're blowups because they're proper birational.
Can anyone see why that would be, say by the fact that the exceptional divisors are chains of $$\mathbb{P}^1$$'s or something?