# 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.

## Definition

Suppose that *Y* is a normal variety such that its canonical class *K*_{Y} is **Q**-Cartier, and let *f*:*X*→*Y* be a resolution of the singularities of *Y*.
Then

where the sum is over the irreducible exceptional divisors, and the *a*_{i} are rational numbers, called the discrepancies.

Then the singularities of *Y* are called:

**terminal**if*a*_{i}> 0 for all*i***canonical**if*a*_{i}≥ 0 for all*i***log terminal**if*a*_{i}> −1 for all*i***log canonical**if*a*_{i}≥ −1 for all*i*.

## Properties

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 *V*^{m} vanishes along any codimension 1 component of the exceptional locus of a resolution of its singularities.

## Classification in small dimensions

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 **C**^{2} by finite subgroups of SL_{2}(**C**).

Two dimensional log terminal singularities are analytically isomorphic to quotients
of **C**^{2} by finite subgroups of GL_{2}(**C**).

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

## Pairs

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.

## References

- 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, ISSN 0003-486X, JSTOR 1971417, MR 924674 - Mori, Shigefumi (1985), "On 3-dimensional terminal singularities",
*Nagoya Mathematical Journal*,**98**: 43–66, ISSN 0027-7630, MR 792770 - 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 605348 - Reid, Miles (1987), "Young person's guide to canonical singularities",
*Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985)*, Proc. Sympos. Pure Math.,**46**, Providence, R.I.: American Mathematical Society, pp. 345–414, MR 927963