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
 * $$\pi: \mathcal{X} \to C,$$

of a variety (or a scheme) to a curve C with origin 0 (e.g., affine or projective line), the fibers
 * $$\pi^{-1}(t)$$

form a family of varieties over C. Then the fiber $$\pi^{-1}(0)$$ may be thought of as the limit of $$\pi^{-1}(t)$$ as $$t \to 0$$. One then says the family $$\pi^{-1}(t), t \ne 0$$ degenerates to the special fiber $$\pi^{-1}(0)$$. The limiting process behaves nicely when $$\pi$$ 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 $$\pi^{-1}(t)$$ is trivial away from a special fiber; i.e., $$\pi^{-1}(t)$$ is independent of $$t \ne 0$$ up to (coherent) isomorphisms, $$\pi^{-1}(t), t \ne 0$$ 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
 * 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.

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 $\pi$: 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.