one obtains $C'$ from $C$ by forgetting the $(V\setminus V')$-marked points and contracting irreducible components until the curve is again stable. The contraction $C'$ inherits a $\hat V'$-marking from $C$, and the $G$-action on $C$ induces a left $(G':=V'\rtimes{{\mathbb{F}}}_q^\times)$-action on $C'$. If $C$ is smooth, then $(C',\lambda',\phi')=(C,\lambda|_{V'},\phi|_{G'})$, and the contraction morphism $C\to C'$ is the identity.

### Grafting {#grafting .unnumbered}

Let $\oV:=V/V'$. Grafting yields a $V$-fern from a given $V'$-fern $(C',\lambda',\phi')$ and $\oV$-fern $(\oC,\overline\lambda,\overline\phi)$. Intuitively, one does this by gluing a copy of $C'$ to each of the $\oV$-marked points of $\oC$. The resulting curve possesses a natural $\hat V$-marking (up to choosing a decomposition $V=V'\oplus U$) and left $G$-action.

### Contraction to the $v$-component {#contraction-to-the-v-component .unnumbered}

This construction associates to $C$ a pair $(C^v,\lambda^v)$ consisting of a ${{\mathbb{P}}}^1$-bundle over $S$, i.e., a scheme that is Zariski-locally isomorphic to ${{\mathbb{P}}}^1_S$,[^2] along with a map $\lambda^v\colon \hat V\to C^v(S)$ and a morphism $C\to C^v$. We call $\lambda^v$ a *$\hat V$-marking* and say $C^v$ is a (smooth) *$\hat V$-marked curve.* The construction is related to the notion of contraction described above. Intuitively, one obtains $C^v$ from $C$ fiberwise by contracting all irreducible components not containing the $v$-marked point. We will mostly be 