  This paper studies the obstructions to deforming a map from a complex variety
to another variety which is an immersion of codimension one. We extend the
classical notion of semiregularity of subvarieties to maps between varieties,
and show that it largely extends the applicability. Then we apply the main
result to several situations. First, we give deformations of non-reduced curves
on surfaces in a geometrically controlled way. Also, we give a simple but
effective criterion for the vanishing of the obstructions to equisingular
deformations of nodal curves on surfaces. Finally, we construct nodal curves
with very small geometric genus on surfaces of general type.
