Canonical singularity

In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety, and terminal singularities are special cases that appear as singularities of minimal models. They were introduced by Reid (1980). Terminal singularities are important in the minimal model program because smooth minimal models do not always exist, and thus one must allow certain singularities, namely the terminal singularities.

DefinitionEdit

Suppose that Y is a normal variety such that its canonical class KY is Q-Cartier, and let f:XY be a resolution of the singularities of Y. Then

 

where the sum is over the irreducible exceptional divisors, and the ai are rational numbers, called the discrepancies.

Then the singularities of Y are called:

terminal if ai > 0 for all i
canonical if ai ≥ 0 for all i
log terminal if ai > −1 for all i
log canonical if ai ≥ −1 for all i.

PropertiesEdit

The singularities of a projective variety V are canonical if the variety is normal, some power of the canonical line bundle of the non-singular part of V extends to a line bundle on V, and V has the same plurigenera as any resolution of its singularities. V has canonical singularities if and only if it is a relative canonical model.

The singularities of a projective variety V are terminal if the variety is normal, some power of the canonical line bundle of the non-singular part of V extends to a line bundle on V, and V the pullback of any section of Vm vanishes along any codimension 1 component of the exceptional locus of a resolution of its singularities.

Classification in small dimensionsEdit

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 Mori (1985).

Two dimensional canonical singularities are the same as du Val singularities, and are analytically isomorphic to quotients of C2 by finite subgroups of SL2(C).

Two dimensional log terminal singularities are analytically isomorphic to quotients of C2 by finite subgroups of GL2(C).

Two dimensional log canonical singularities have been classified by Kawamata (1988).

PairsEdit

More generally one can define these concepts for a pair   where   is a formal linear combination of prime divisors with rational coefficients such that   is  -Cartier. The pair is called

  • terminal if Discrep 
  • canonical if Discrep 
  • klt (Kawamata log terminal) if Discrep  and  
  • plt (purely log terminal) if Discrep 
  • lc (log canonical) if Discrep .

ReferencesEdit

  • Kollár, János (1989), "Minimal models of algebraic threefolds: Mori's program", Astérisque (177): 303–326, ISSN 0303-1179, MR 1040578
  • Kawamata, Yujiro (1988), "Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces", Ann. of Math., 2, 127 (1): 93–163, doi:10.2307/1971417, ISSN 0003-486X, JSTOR 1971417, MR 0924674
  • Mori, Shigefumi (1985), "On 3-dimensional terminal singularities", Nagoya Mathematical Journal, 98: 43–66, doi:10.1017/s0027763000021358, ISSN 0027-7630, MR 0792770
  • Reid, Miles (1980), "Canonical 3-folds", Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Alphen aan den Rijn: Sijthoff & Noordhoff, pp. 273–310, MR 0605348
  • Reid, Miles (1987), "Young person's guide to canonical singularities", Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Providence, R.I.: American Mathematical Society, pp. 345–414, MR 0927963