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In this talk, we will consider deformations of singular complex curves on complex
surfaces. More precisely, if $\varphi\colon C\to S$ is a map from a smooth projective
curve to a projective surface, we consider the deformation of $\varphi$. Despite the
simplicity of the problem, little seems to be known for surfaces of positive Kodaira
dimension. The problem of the existence of deformations can be reduced to two more
tractable problems: checking certain cohomological condition, and solving a certain
system of polynomial equations which is independent of geometry. The latter problem is
almost always expected to be solved, and in this case, the map has virtually optimal
deformation property. This talk will be based on arxiv:2310.14039.
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