388:
Two dimensional terminal singularities are smooth. If a variety has terminal singularities, then its singular points have codimension at least 3, and in particular in dimensions 1 and 2 all terminal singularities are smooth. In 3 dimensions they are isolated and were classified by
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because smooth minimal models do not always exist, and thus one must allow certain singularities, namely the terminal singularities.
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852:(1988), "Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces",
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Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic
Geometry, Angers, 1979
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415:
Two dimensional log terminal singularities are analytically isomorphic to quotients of
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is a formal linear combination of prime divisors with rational coefficients such that
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Two dimensional log canonical singularities have been classified by
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where the sum is over the irreducible exceptional divisors, and the
182:{\displaystyle \displaystyle K_{X}=f^{*}(K_{Y})+\sum _{i}a_{i}E_{i}}
948:, Alphen aan den Rijn: Sijthoff & Noordhoff, pp. 273–310,
967:, Proc. Sympos. Pure Math., vol. 46, Providence, R.I.:
963:(1987), "Young person's guide to canonical singularities",
965:
Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985)
823:"Minimal models of algebraic threefolds: Mori's program"
442:
More generally one can define these concepts for a pair
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Two dimensional canonical singularities are the same as
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appear as singularities of the canonical model of a
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400:, and are analytically isomorphic to quotients of
372:vanishes along any codimension 1 component of the
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66:is a normal variety such that its canonical class
42:are special cases that appear as singularities of
50:. Terminal singularities are important in the
708:{\displaystyle \lfloor \Delta \rfloor \leq 0}
8:
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344:The singularities of a projective variety
301:The singularities of a projective variety
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901:"On 3-dimensional terminal singularities"
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91:be a resolution of the singularities of
292:multiplier ideal (algebraic geometry)
27:Singularities of projective varieties
7:
390:
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384:Classification in small dimensions
800:{\displaystyle (X,\Delta )\geq -1}
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642:(Kawamata log terminal) if Discrep
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25:
754:{\displaystyle (X,\Delta )>-1}
676:{\displaystyle (X,\Delta )>-1}
630:{\displaystyle (X,\Delta )\geq 0}
201:are rational numbers, called the
720:(purely log terminal) if Discrep
587:{\displaystyle (X,\Delta )>0}
305:are canonical if the variety is
368:the pullback of any section of
348:are terminal if the variety is
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969:American Mathematical Society
944:(1980), "Canonical 3-folds",
549:-Cartier. The pair is called
520:{\displaystyle K_{X}+\Delta }
542:{\displaystyle \mathbb {Q} }
360:extends to a line bundle on
356:of the non-singular part of
317:extends to a line bundle on
313:of the non-singular part of
905:Nagoya Mathematical Journal
467:{\displaystyle (X,\Delta )}
1014:
766:(log canonical) if Discrep
289:
208:Then the singularities of
46:. They were introduced by
918:10.1017/s0027763000021358
419:by finite subgroups of GL
404:by finite subgroups of SL
282:≥ −1 for all
339:relative canonical model
487:{\displaystyle \Delta }
32:canonical singularities
18:Canonical singularities
801:
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380:of its singularities.
333:of its singularities.
264:> −1 for all
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40:terminal singularities
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354:canonical line bundle
311:canonical line bundle
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52:minimal model program
971:, pp. 345–414,
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398:du Val singularities
352:, some power of the
309:, some power of the
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998:Algebraic geometry
993:Singularity theory
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246:≥ 0 for all
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79:-Cartier, and let
36:projective variety
374:exceptional locus
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16:(Redirected from
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850:Kawamata, Yujiro
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829:(177): 303–326,
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30:In mathematics,
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897:Mori, Shigefumi
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866:10.2307/1971417
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228:> 0 for all
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44:minimal models
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860:(1): 93–163,
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854:Ann. of Math.
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819:Kollár, János
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325:has the same
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271:log canonical
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203:discrepancies
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62:Suppose that
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253:log terminal
252:
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229:
224:
220:
216:
212:are called:
209:
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63:
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39:
31:
29:
961:Reid, Miles
942:Reid, Miles
391:Mori (1985)
327:plurigenera
48:Reid (1980)
987:Categories
827:Astérisque
812:References
599:if Discrep
556:if Discrep
378:resolution
331:resolution
297:Properties
290:See also:
58:Definition
927:0027-7630
911:: 43–66,
874:0003-486X
835:0303-1179
792:−
789:≥
783:Δ
746:−
737:Δ
700:≤
697:⌋
694:Δ
691:⌊
668:−
659:Δ
622:≥
616:Δ
597:canonical
573:Δ
515:Δ
482:Δ
459:Δ
235:canonical
150:∑
125:∗
899:(1985),
821:(1989),
554:terminal
217:terminal
95:. Then
977:0927963
954:0605348
935:0792770
890:0924674
882:1971417
843:1040578
329:as any
975:
952:
933:
925:
888:
880:
872:
841:
833:
474:where
364:, and
350:normal
321:, and
307:normal
38:, and
878:JSTOR
856:, 2,
438:Pairs
376:of a
923:ISSN
870:ISSN
831:ISSN
743:>
683:and
665:>
579:>
273:if
255:if
237:if
219:if
913:doi
862:doi
858:127
718:plt
640:klt
527:is
427:).
412:).
205:.
75:is
989::
973:MR
950:MR
931:MR
929:,
921:,
909:98
907:,
903:,
886:MR
884:,
876:,
868:,
839:MR
837:,
825:,
764:lc
434:.
393:.
341:.
915::
864::
807:.
795:1
786:)
780:,
777:X
774:(
749:1
740:)
734:,
731:X
728:(
703:0
671:1
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656:,
653:X
650:(
625:0
619:)
613:,
610:X
607:(
582:0
576:)
570:,
567:X
564:(
536:Q
512:+
507:X
503:K
462:)
456:,
453:X
450:(
425:C
423:(
421:2
417:C
410:C
408:(
406:2
402:C
370:V
366:V
362:V
358:V
346:V
335:V
323:V
319:V
315:V
303:V
286:.
284:i
279:i
275:a
266:i
261:i
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248:i
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239:a
230:i
225:i
221:a
210:Y
198:i
194:a
174:i
170:E
164:i
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146:+
143:)
138:Y
134:K
130:(
121:f
117:=
112:X
108:K
93:Y
89:Y
87:→
85:X
83::
81:f
77:Q
72:Y
68:K
64:Y
20:)
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