interested in the case $v=\infty$, but also consider the case $v=0$. When $C$ is smooth, we obtain $C^v$ by simply forgetting the $G$-action on $C$.

### Line bundles {#line-bundles .unnumbered}

We also provide constructions associating line bundles with certain extra structure to $(C,\lambda,\phi)$. One such construction assigns to $C$ a pair $(L,\lambda)$ consisting of a line bundle $L$ over $S$ along with a fiberwise non-zero (see Definition \[fiberwise\]) ${{\mathbb{F}}}_q$-linear map $\lambda\colon V\to L(S).$ To obtain $L$, we take the contraction to the $\infty$-component $C^\infty$ and define $L$ to be the complement of the image of the $\infty$-section.

Let $\mrV:=V\setminus\{0\}.$ An analogous construction yields a pair $(\hat L,\rho)$ consisting of a line bundle $\hat L$ over $S$ and a fiberwise non-zero reciprocal map (see Definition \[recipmap\]) $\rho\colon \mrV\to \hat L(S).$ This construction differs from the preceding one in that we instead take the contraction to the $0$-component $C^0$ and define $\hat L$ to be the complement of the $0$-marked point.

Main theorem and outline of the proof
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Our main result is the following:

The scheme $B_V$ represents the functor that associates an ${{\mathbb{F}}}_q$-scheme $S$ to the set of isomorphism classes of $V$-ferns over $S$.

We denote the functor referred to 