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In algebraic geometry, a degeneration is the act of taking a limit of a family of varieties. Precisely, given a morphism
of a variety to a curve C with origin 0 , the fibers
form a family of varieties over C. Then the fiber π − 1 {\displaystyle \pi ^{-1}} may be thought of as the limit of π − 1 {\displaystyle \pi ^{-1}} as t → 0 {\displaystyle t\to 0}. One then says the family π − 1 , t ≠ 0 {\displaystyle \pi ^{-1},t\neq 0} degenerates to the special fiber π − 1 {\displaystyle \pi ^{-1}}. The limiting process behaves nicely when π {\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 π − 1 {\displaystyle \pi ^{-1}} is trivial away from a special fiber; i.e., π − 1 {\displaystyle \pi ^{-1}} is independent of t ≠ 0 {\displaystyle t\neq 0} up to isomorphisms, π − 1 , t ≠ 0 {\displaystyle \pi ^{-1},t\neq 0} is called a general fiber.