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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.
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over a discrete valuation ring. If the generic fiber is ruled, then each irreducible component of the special fiber is also ruled.
814:
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567:, the important point is to understand the boundaries of the moduli, which amounts to understand degenerations of curves.
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V. Alexeev, Ch. Birkenhake, and K. Hulek, Degenerations of Prym varieties, J. Reine Angew. Math. 553 (2002), 73–116.
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Vanishing, Singularities And
Effective Bounds Via Prime Characteristic Local Algebra.
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35:) is the act of taking a limit of a family of varieties. Precisely, given a morphism
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701:) are affine, then an embedded infinitesimal deformation amounts to an ideal
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if it is flat and the fiber of it over the distinguished point 0 of
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Ruled-ness specializes. Precisely, Matsusaka'a theorem says
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with origin 0 (e.g., affine or projective line), the fibers
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362:. Many authors assume degenerations to be flat.
358:and, in that case, the degeneration is called a
845:– Tata Institute of Fundamental Research, 1976
634:embedded first-order infinitesimal deformation
8:
404:is trivial away from a special fiber; i.e.,
862:, vol. 52, New York: Springer-Verlag,
334:. The limiting process behaves nicely when
843:Lectures on Deformations of Singularities
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70:{\displaystyle \pi :{\mathcal {X}}\to C,}
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800:and there is some choice of embedding.
80:of a variety (or a scheme) to a curve
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893:An invitation to toric degenerations
753:In general, given a pointed scheme (
513:{\displaystyle \pi ^{-1}(t),t\neq 0}
281:{\displaystyle \pi ^{-1}(t),t\neq 0}
830:Relative effective Cartier divisor
172:may be thought of as the limit of
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886:Deformations of algebraic schemes
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129:form a family of varieties over
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860:Graduate Texts in Mathematics
620:a scheme of finite type over
436:{\displaystyle \pi ^{-1}(t)}
397:{\displaystyle \pi ^{-1}(t)}
327:{\displaystyle \pi ^{-1}(0)}
204:{\displaystyle \pi ^{-1}(t)}
165:{\displaystyle \pi ^{-1}(0)}
119:{\displaystyle \pi ^{-1}(t)}
663:) such that the projection
624:. Given a closed subscheme
520:is called a general fiber.
237:. One then says the family
948:
596:Infinitesimal deformations
27:In algebraic geometry, a
761:, a morphism of schemes
571:Stability of invariants
524:Degenerations of curves
462:{\displaystyle t\neq 0}
891:M. Gross, M. Siebert,
678:as the special fiber.
640:is a closed subscheme
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230:{\displaystyle t\to 0}
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347:{\displaystyle \pi }
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825:Frobenius splitting
820:Kodaira–Spencer map
932:Algebraic geometry
855:Algebraic Geometry
810:deformation theory
757:, 0) and a scheme
543:. You can help by
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869:978-0-387-90244-9
850:Hartshorne, Robin
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133:. Then the fiber
18:Flat degeneration
16:(Redirected from
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552:November 2019
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539:This section
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356:flat morphism
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883:E. Sernesi:
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780:of a scheme
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670:→ Spec
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587:irreducible
580:
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549:
545:adding to it
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292:
288:
130:
128:
81:
79:
32:
29:degeneration
28:
26:
778:deformation
289:degenerates
841:M. Artin,
836:References
712:such that
505:≠
482:−
478:π
454:≠
417:−
413:π
378:−
374:π
342:π
308:−
304:π
273:≠
250:−
246:π
222:→
185:−
181:π
146:−
142:π
100:−
96:π
59:→
46:π
926:Category
852:(1977),
804:See also
878:0463157
796:= Spec
693:= Spec(
685:= Spec
608:be the
293:special
291:to the
876:
866:
585:normal
295:fiber
769:'
731:'
720:'
705:'
667:'
659:Spec(
653:Spec(
644:'
583:be a
354:is a
864:ISBN
689:and
616:and
600:Let
579:Let
31:(or
788:is
746:is
734:in
708:of
681:If
647:of
636:of
628:of
547:.
211:as
928::
874:MR
872:,
858:,
772:→
765::
750:.
738:=
716:/
604:=
798:D
794:S
790:X
786:S
782:X
774:S
767:X
763:π
759:X
755:S
748:I
744:ε
742:/
740:A
736:A
729:I
725:D
718:I
714:A
710:A
703:I
699:I
697:/
695:A
691:X
687:A
683:Y
676:X
672:D
665:X
661:D
657:)
655:k
651:×
649:Y
642:X
638:X
630:Y
626:X
622:k
618:Y
614:k
606:k
602:D
581:X
554:)
550:(
508:0
502:t
499:,
496:)
493:t
490:(
485:1
457:0
451:t
431:)
428:t
425:(
420:1
392:)
389:t
386:(
381:1
322:)
319:0
316:(
311:1
276:0
270:t
267:,
264:)
261:t
258:(
253:1
225:0
219:t
199:)
196:t
193:(
188:1
160:)
157:0
154:(
149:1
131:C
114:)
111:t
108:(
103:1
82:C
65:,
62:C
54:X
49::
20:)
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