Obstructions to deforming maps from curves to surfaces
This paper studies the obstruction to deforming a map from a complex curve to a complex surface, under the condition that the map is locally an embedding. We extend the classical notion of semiregularity of embedded curves to maps, and show that it largely extends the applicability. As an example, we apply it to K3 surfaces, and show that a generic complex polarized K3 surface contains infinitely many $g$ dimensional families of curves of geometric genus $g$ for any positive integer $g$.