this shows that we may specialize a morphism in ${\overline { {\mathcal M}} _{bir} \bigl( X, \beta \bigr) }$ so that its image is a union of lines. Second, we need explicit geometric arguments to deal with the low degree cases. Again, in the case of the projective plane, this step is used to bring the domain to a standard form (a chain of rational curves, rather than a general rational tree), while preserving the property that the image of the morphism consists of a union of lines.

To analyze the curves of low anticanonical degree on a del Pezzo surface, we need a detailed description of their divisor classes in ${\rm Pic} (X)$. In particular, we use the group of symmetries of the Picard lattice to reduce the number of cases to treat. Section \[sporco\] is devoted to this analysis.

Two technical deformation-theoretic tools prove useful. The first is a description of the obstruction space of a stable map to a smooth surface in terms of combinatorial invariants of the map. This is proved in Section \[conormale\]. The second is a lifting result that allows us, given a deformation of a component of a curve, to get a deformation 