Degeneration (algebraic geometry)

In algebraic geometry, a degeneration is the act of taking a limit of a family of varieties. Precisely, given a morphism
of a variety to a curve C with origin 0, the fibers
form a family of varieties over C. Then the fiber may be thought of as the limit of as. One then says the family degenerates to the special 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.
When the family is trivial away from a special fiber; i.e., is independent of up to isomorphisms, is called a general fiber.

Degenerations of curves

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.

Stability of invariants

Ruled-ness specializes. Precisely, Matsusaka'a theorem says

Infinitesimal deformations

Let D = k 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 Spec such that the projection is flat and has X as the special fiber.
If Y = Spec A and are affine, then an embedded infinitesimal deformation amounts to an ideal of A such that is flat over D and the image of in A = A/''ε is.
In general, given a pointed scheme and a scheme X'', a morphism of schemes : X → S is called the deformation of a scheme X if it is flat and the fiber of it over the distinguished point 0 of S is X. Thus, the above notion is a special case when S = Spec D and there is some choice of embedding.