of the whole curve. The statement is proved in Lemma \[ovvio\] and the construction following it is the way in which we are typically going to use it. This is specific to the surface case. The lifting result allows us to deform a map with reducible domain by deforming only a few components at a time. This is done systematically in Section \[rompisezione\]. We are therefore able to reduce the general problem to relatively few special cases, cf. Theorem \[passo\]. The explicit computation of the obstruction spaces allows us to prove that in the deformations performed we never move to a different irreducible component of the moduli space.

The connectedness of the locus of morphisms in standard form is a consequence of some explicit computations, some of which are reformulations of classical geometric statements, such as the fact that the ramification locus of the projection from a general point on a smooth cubic surface in ${\mathbb P}^3$ is a smooth plane quartic curve. This is the content of Section \[pochipochi\], but see also Section \[conichette\].

Cohomology Groups and Obstruction Spaces
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Rational Trees
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The purpose of this section is to prove some general results which are useful to compute 