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

${\displaystyle \pi$ :{\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.