morphism $X\to S$ and $s\in S$, we denote the fiber over $s$ by $X_s$. Given another morphism $f\colon T\to S$, we often denote the base change $X\times_S T$ by $f^*X$. For an open immersion $j\colon U\into S$, we will usually write $X|_U$ in place of $j^*X$.

We always view the irreducible components of a scheme as closed subschemes endowed with the induced reduced scheme structure.

Stable marked curves of genus 0 {#stablemarkedcurves}
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In this section we collect some useful facts and constructions involving stable marked curves of genus 0. The main references are [@Arb] and [@Knud]. Let $S$ be an arbitrary scheme, and let $I$ be a finite set.

Stable $I$-marked curves of genus 0
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\[stablecurve\] An *$I$-marked curve* $(C,\lambda)$ of genus 0 over $S$ is a flat and proper scheme $C$ over $S$, together with a map $$\lambda\colon I\to C(S),\, i\mapsto \lambda_i$$ such that for every geometric point $\os$ of $S$,

1.  the geometric fiber $C_\os$ is a reduced and connected curve with at most ordinary double point singularities,

2.  the equality $\dim H^1(C_\os,{{\cal O}}_{C_\os})=0$ holds.

A closed point $p\in C_{\os}$ is called *marked* if $p=\lambda_i(\os)$ for some $i\in I$. The point $p$ is called *special* if 