Let $Y$ be a normal variety such that $K_Y$ is $\mathbb{Q}$-Cartier, and $f:X\to Y$ a resolution of singularities. Then, $$ K_X = f^\ast(K_Y) +\sum_i a_iE_i $$ where $a_i \in \mathbb{Q}$ and the $E_i$ are the irreducible exceptional divisors. Then the singularities of $Y$ are terminal, canonical, log terminal or log canonical if $a_i > 0, \geq 0, >-1$ or $\geq -1$, respectively.

Question: Using this definition, can you give some examples of log terminal singularities that are not canonical and log canonical singularities that are not log terminal?

From Reid's "Young Peron's Guide to Canonical Singularities" I have nontrivial examples of the first two types:

A canonical singularity: Let $Y = \lbrace xz=y^2 \rbrace \subset \mathbb{A}^3$. There is an ordinary double point at the origin, and when we blow up we get a (-2)-curve $E$. By adjunction, $K_X = f^\ast K_Y$, i.e., $a = 0$, so by definition, $Y$ has a canonical singularity.

Terminal Singularities: The cone over the Veronese surface has a resolution $X$ with exceptional locus $E\simeq \mathbb{P}^2$. One can show (adjunction again) $K_X = f^\ast K_Y + \frac{1}{2}E$, and so the singularities are terminal but not canonical.


1 Answer 1


The book Birational Geometry of Algebraic Varieties by Kollár--Mori goes into these in some detail.

So for example they mention (Proposition 4.18) that canonical surface singularities (= du Val singularities) are (locally analytically) quotients of $\mathbf{C}^2$ by finite subgroups of $SL(2,\mathbf{C})$, whereas log terminal surface singularities are quotients by arbirtrary finite groups.

As for the other question, log terminal singularities are always rational. So it suffices to find a log canonical but not rational singularity: the classic example is the cone over an elliptic curve.

The recent book Singularities of the Minimal Model Program by Kollár--Kovács discusses these things in much greater detail, but I must admit I haven't read it.


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