Canonical singularity

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

${\displaystyle \displaystyle K_{X}=f^{*}(K_{Y})+\sum _{i}a_{i}E_{i}}$

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.

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.

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).

More generally one can define these concepts for a pair ${\displaystyle (X,\Delta )}$ where ${\displaystyle \Delta }$ is a formal linear combination of prime divisors with rational coefficients such that ${\displaystyle K_{X}+\Delta }$ is ${\displaystyle \mathbb {Q} }$-Cartier. The pair is called

• terminal if Discrep${\displaystyle (X,\Delta )>0}$
• canonical if Discrep${\displaystyle (X,\Delta )\geq 0}$
• klt (Kawamata log terminal) if Discrep${\displaystyle (X,\Delta )>-1}$ and ${\displaystyle \lfloor \Delta \rfloor \leq 0}$
• plt (purely log terminal) if Discrep${\displaystyle (X,\Delta )>-1}$
• lc (log canonical) if Discrep${\displaystyle (X,\Delta )\geq -1}$.

• 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