given by the base change of $Q_V$ to $\Spec {{\mathbb{F}}}_q(t)$. This reduction is mainly accomplished using the following two observations. First, if $I'\subset I\subset A$ are proper non-zero ideals, then $\overline{M}^n_{A,I}$ can be realized as a quotient of $\overline{M}^n_{A,I'}$ under the action of a finite group. Second, for certain $A\subset A'$ and $I':=IA'$ and $n=n'\cdot[F'/F]$, where $F$ and $F'$ are the corresponding quotient fields, there is a finite injective morphism $M^{n'}_{A',I'}\to M^{n}_{A,I}$. Then $\overline{M}^{n'}_{A',I'}$ is the normalization of $\overline{M}^{n}_{A,I}$ in the function field of $M^{n'}_{A',I'}$. In proving the existence of the Satake compactification from the base case, the first observation allows one to increase the level, and the second allows one to enlarge the admissible coeffecient ring.

One might use similar reasoning to generalize $B_V$. First, appropriately generalize the notion of a $V$-fern and consider the corresponding moduli functor. Then exploit the morphisms $M^n_{I',A}\to M^n_{I,A}$ and $M^{n'}_{I',A'}\to M^n_{A,I}$ to reduce the question of representabiliy to the case $A={{\mathbb{F}}}_q[t]$ and $I=(t)$. It is our hope that the foundations laid here will lead to the success of such an approach.

Outline {#outline .unnumbered}
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**Section 1.** We review the notion of *stable marked curves* of genus 0 and recall the concepts of 