though the correspondence with smooth $V$-ferns then breaks down.

General $V$-ferns
-----------------

To motivate the general definition of a $V$-fern, let $R$ be a discrete valuation ring over ${{\mathbb{F}}}_q$, and suppose we are given a smooth $V$-fern $(C,\lambda,\phi)$ over the generic point of $\Spec R$. Knudsen shows in [@Knud] that there is a fine moduli space ${{\cal M}}_{\hat V}$ of stable $\hat V$-marked curves and that ${{\cal M}}_{\hat V}$ is proper. The valuative criterion for properness ([@Ha §II.4]) then says that the morphism $\Spec K\to {{\cal M}}_{\hat V}$ corresponding to $(C,\lambda)$ extends uniquely to a morphism $\Spec R\to {{\cal M}}_{\hat V}$. On other words, the curve $(C,\lambda)$ extends uniquely to a stable $\hat V$-marked curve $(\tilde C,\tilde\lambda)$ over $\Spec R$. By the functoriality of the extension, the $G$-action $\phi$ can be extended uniquely to $\tilde C$. The corresponding tuple $(\tilde C,\tilde\lambda,\tilde\phi)$ is characterized by certain conditions generalizing those in Definition \[smoothferndef\]. It is important to note that the special fiber of $\tilde C$ can be singular, and hence the resulting object is **not** in general a smooth $V$-fern over $\Spec R$.

It is then natural to define a $V$-fern over a general ${{\mathbb{F}}}_q$-scheme $S$ as a tuple $(C,\lambda,\phi)$, where $(C,\lambda)$ 