In algebraic geometry, a degeneration (or specialization) is the act of taking a limit of a family of varieties. Precisely, given a morphism
\pi:l{X}\toC,
\pi-1(t)
\pi-1(0)
\pi-1(t)
t\to0
\pi-1(t),t\ne0
\pi-1(0)
\pi
When the family
\pi-1(t)
\pi-1(t)
t\ne0
\pi-1(t),t\ne0
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.
Ruled-ness specializes. Precisely, Matsusaka'a theorem says
Let X be a normal irreducible projective scheme over a discrete valuation ring. If the generic fiber is ruled, then each irreducible component of the special fiber is also ruled.
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.