Canonical singularity

In mathematics, canonical singularities are a class of singularities that appear on the canonical model of an algebraic variety, and terminal singularities are a narrower class that occur as singularities of minimal models. These classes of singularities were introduced by Miles Reid (1980). Terminal singularities are important in the minimal model program because smooth minimal models do not exist in the desired generality, and hence certain "mild" singularities must be allowed.

Let X be a normal variety over a field whose canonical class K_X is ${\displaystyle \mathbb {Q} }$-Cartier (as discussed below), and let ${\displaystyle f\colon Y\to X}$ be a resolution of singularities of X. Using that Cartier divisors can be pulled back, one can write

where the sum is over the exceptional divisors of f (the codimension-1 subvarieties of Y, these being irreducible by definition, whose image in X has codimension at least 2). The a_i are rational numbers, called the discrepancies.

Then X is said to be

(One can also say that X has "terminal singularities" or "canonical singularities".) These properties are independent of the choice of resolution.

Suppose, more strongly, that ${\displaystyle f\colon Y\to X}$ is a log resolution, meaning that Y is nonsingular and the exceptional locus of f is a divisor with simple normal crossings in Y. Then X is said to be

These two properties are independent of the choice of log resolution. They were introduced in the early 1980s (with slightly different terminology) by Yujiro Kawamata.

If some log resolution of X has an exceptional divisor with discrepancy ${\displaystyle a_{i}}$ less than ${\displaystyle -1}$, then X has other log resolutions with arbitrarily negative discrepancies ${\displaystyle a_{j}}$. As a result, "log canonical" is the most general condition that can be defined along these lines, independent of the choice of log resolution.