is a stable $\hat V$-marked curve over $S$ and $\phi\colon G\to\Aut_S(C)$ is a left group action of $G$ on $C$ satisfying certain conditions generalizing those of Definition \[smoothferndef\] (see Definition \[FernDef\]). Note that we often write $C$ in place of $(C,\lambda,\phi)$. A smooth $V$-fern is then just a $V$-fern with smooth fibers. Since this is an open condition, Proposition \[smoothferns0\] implies that the moduli space of $V$-ferns, if it exists, contains $\OV$ as an open subscheme. The argument that a smooth $V$-fern over the generic point of $\Spec R$ extends uniquely to a $V$-fern over $\Spec R$ applies just as well in the case of a general $V$-fern over the generic point. The moduli space of $V$-ferns would thus satisfy the valuative criterion for properness, so the notion of $V$-ferns points toward a modular compactification of $\OV$.

Constructions involving $V$-ferns
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We will use several constructions involving $V$-ferns repeatedly and in combination. For convenience, we briefly introduce each of them here. Let $S$ be a scheme over ${{\mathbb{F}}}_q$ and let $(C,\lambda,\phi)$ be a $V$-fern over $S$. Consider a subspace $0\neq V'\subsetneq V.$

### Contraction {#contraction .unnumbered}

Contraction associates a $V'$-fern $(C',\lambda',\phi')$ to $C$, along with a morphism $C\to C'$. Conceptually, 