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

π:𝒳->C,

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

π−1(t)

form a family of varieties over C. Then the fiber π−1(0) may be thought of as the limit of π−1(t) as t->0. One then says the family π−1(t),t≠0 degenerates to the special fiber π−1(0). The limiting process behaves nicely when π 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(t) is trivial away from a special fiber; i.e., π−1(t) is independent of t≠0 up to (coherent) isomorphisms, π−1(t),t≠0 is called a general fiber.

## Contents

* 1 Degenerations of curves
* 2 Stability of invariants
* 3 Infinitesimal deformations
* 4 See also
* 5 References
* 6 External links

## 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
* 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.

## External links

* http://mathoverflow.net/questions/88552/when-do-infinitesimal-deformations-lift-to-global-deformations

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