“Draft:Degeneration (algebraic geometry)”的意思、由来-开放百科全书

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Draft:Degeneration (algebraic geometry)

释义

Definitions
Flat degeneration Infinitesimal deformations
Examples
Pencils Degenerations of curves Toric degenerations
Principle of continuity
Stability of invariants
See also
References
External links

(It is possible that we might want separate articles for the infinitesimal situation and the global situation.)

TODO: write the lede when the draft is more complete.

Definitions

Flat degeneration

A flat degeneration is a flat morphism

The fiber over the zero is called the special fiber or the zero fiber.

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.

Examples

Pencils

A pencil is a one-parameter family of divisors.

Degenerations of curves

TODO: use http://projecteuclid.org/euclid.bams/euclid.bams/1183551576

Adapt examples from https://arxiv.org/pdf/1207.1048.pdf

Toric degenerations

A toric degeneration is a flat degeneration when the special fiber is a not-necessary-normal toric variety.

Principle of continuity

The principle of continuity states, roughly, that intersection-theoretic quantities are preserved under degenerations.

For general properties of flat morphisms, see flat module as well as flat morphism. The present article focuses on general techniques.

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.

References

M. Artin, Lectures on Deformations of Singularities - Tata Institute of Fundamental Research, 1976
{{Hartshorne AG}}
E. Sernesi: [https://books.google.com/books/about/Deformations_of_algebraic_schemes.html?id=xkcpQo9tBN8C&hl=en Deformations of algebraic schemes]
M. Gross, M. Siebert, [https://arxiv.org/pdf/0808.2749.pdf 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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