happen that $R(\beta ) = \emptyset $, while $\overline {\mathcal M}_{0,0} \bigl( X, \beta \bigr) \neq \emptyset $.

The mapping spaces parametrize the set of all (stable) maps to the surface $X$ from possibly reducible curves. The domain curves of the maps in $\overline {\mathcal M}_{0,0} \bigl( X, \beta \bigr)$ are connected and nodal, have all components isomorphic to ${\mathbb P}^1$ and the components are attached in such a way that the resulting topological space is simply connected. We refer to such a domain curve as a “rational tree.” Taking the image of a map yields a morphism $FC$ from (the semi-normalization of) $\overline {\mathcal M}_{0,0} \bigl( X, \beta \bigr)$ to the closure in $|\beta |$ of $R(\beta )$ (see [@Ko] Section I.6).

Let ${\overline { {\mathcal M}} _{bir} \bigl( X, \beta \bigr) }$ be the closure of the subspace of ${\overline { {\mathcal M}} _{0,0} \bigl( X, \beta \bigr)}$ consisting of morphisms $f : C \rightarrow X$, with $C \simeq {\mathbb P}^1$ and $f$ birational onto its image. In many interesting cases it is true and easy to check that the map $FC$ defined above is in fact birational, when restricted to ${\overline { {\mathcal M}} _{bir} \bigl( X, 