it is singular or marked. We say that $(C,\lambda)$ is *stable* if, in addition to the above,

3.  the curve $C_\os$ is smooth at the marked points $\lambda_i(\os)$,

4.  we have $\lambda_i(\os)\neq \lambda_j(\os)$ for all $i\neq j$,

5.  each irreducible component of $C_\os$, contains at least 3 special points.

Conditions (1) and (2) imply that each geometric fiber is a tree of copies of ${{\mathbb{P}}}^1$. We will see later that this already holds on (scheme-theoretic) fibers (Proposition \[P1fibs\]).

We call the map $\lambda\colon I\to C(S)$ an *$I$-marking* on $C$. From now on, we abbreviate the expression “$I$-marked curve of genus 0” by $I$-marked curve. We will often write $C$ for the pair $(C,\lambda)$ if confusion is unlikely.

In [@Knud] a morphism of stable $I$-marked curves $f\colon (C,\lambda)\to(D,\mu)$ over a scheme $S$ is defined to be an isomorphism over $S$ such that $f\circ\lambda_i=\mu_i$ for all $i\in I$. In the following paragraphs, we will define a more general notion of morphisms between (not necessarily stable) $I$-marked curves that is equivalent to Knudsen’s definition when the curves are stable. One of the main results in [@Knud], written here in the context of stable $I$-marked curves is the following:

\[moduli\]The functor ${{\cal M}}_I$ associating 