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.

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:

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).

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