In algebraic geometry, a degeneration (or specialization) is the act of taking a limit of a family of varieties.
Precisely, given a morphism of a variety (or a scheme) to a curve C with origin 0 (e.g., affine or projective line), the fibers form a family of varieties over C. Then the fiber
The limiting process behaves nicely when
is a flat morphism and, in that case, the degeneration is called a flat degeneration.
Many authors assume degenerations to be flat.
is trivial away from a special fiber; i.e.,
is called a general fiber.
In the study of moduli of curves, the important point is to understand the boundaries of the moduli, which amounts to understand degenerations of curves.
Precisely, Matsusaka'a theorem says Let D = k[ε] be the ring of dual numbers over a field k and Y a scheme of finite type over k. Given a closed subscheme X of Y, by definition, an embedded first-order infinitesimal deformation of X is a closed subscheme X' of Y ×Spec(k) Spec(D) such that the projection X' → Spec D is flat and has X as the special fiber.
If Y = Spec A and X = Spec(A/I) are affine, then an embedded infinitesimal deformation amounts to an ideal I' of A[ε] such that A[ε]/ I' is flat over D and the image of I' in A = A[ε]/ε is I.
Thus, the above notion is a special case when S = Spec D and there is some choice of embedding.