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Degeneration (algebraic geometry)

## Summary

In algebraic geometry, a degeneration (or specialization) is the act of taking a limit of a family of varieties. Precisely, given a morphism

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

of a variety (or a scheme) to a curve C with origin 0 (e.g., affine or projective line), the fibers

${\displaystyle \pi ^{-1}(t)}$

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