Consider a flat family of projective curves (not necessarily embedded). Assume the
generic fibre is a smooth (connected) curve of genus g. What are the possible
singularities of the central fibre?
The analogous local question ("what are the possible degenerations of a smooth
curvegerm?") is meaningless.
But for the projective curves much is controlled by global geometry.
Degenerations of curves arise in various questions, e.g.
in the study of surfaces, when trying to present them as fibrations. Degenerations of
P^1 are known since 19'th century, the case g=1 was done by Kodaira. Many things are
known for small genera.
This is a general introduction to the subject.
