Degeneration (algebraic geometry)

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 formula_3 may be thought of as the limit of formula_2 as formula_5. One then says the family formula_6 "degenerates" to the "special" fiber formula_3. The limiting process behaves nicely when formula_8 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 formula_2 is trivial away from a special fiber; i.e., formula_2 is independent of formula_11 up to (coherent) isomorphisms, formula_6 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"["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".
In general, given a pointed scheme ("S", 0) 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.