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

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

form a family of varieties over *C*. Then the fiber may be thought of as the limit of as . One then says the family *degenerates* to the *special* fiber . The limiting process behaves nicely when 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 is trivial away from a special fiber; i.e., is independent of up to (coherent) isomorphisms, 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.

## See also

- deformation theory
- differential graded Lie algebra
- Kodaira–Spencer map
- Frobenius splitting
- Relative effective Cartier divisor

## References

- M. Artin, Lectures on Deformations of Singularities – Tata Institute of Fundamental Research, 1976
- Hartshorne, Robin (1977),
*Algebraic Geometry*, Graduate Texts in Mathematics,**52**, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 - E. Sernesi:
*Deformations of algebraic schemes* - M. Gross, M. Siebert, An invitation to toric degenerations
- M. Kontsevich, Y. Soibelman: Affine structures and non-Archimedean analytic spaces, in: The unity of mathematics (P. Etingof, V. Retakh, I.M. Singer, eds.), 321–385, Progr. Math. 244, Birkh ̈auser 2006.
- Karen E Smith,
*Vanishing, Singularities And Effective Bounds Via Prime Characteristic Local Algebra.* - V. Alexeev, Ch. Birkenhake, and K. Hulek, Degenerations of Prym varieties, J. Reine Angew. Math. 553 (2002), 73–116.

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