In the Corollary at pag 407 of Young persons guide to canonical singularities there is a formula to compute the contributions $c\_q(D)$ to Riemann-Roch of a divisor $D$ passing through a point $q\in X$, where $X$ is a normal surface, $q\in X$ is a quotient singularity of type $\frac{1}{r}(a\_1,a\_2)$, and $L = \mathcal{O}\_X(D)$ is of type $i(\frac{1}{r}(a\_1,a\_2))$ at $q$ for $i = 0,...,r-1$.


Now, let us take the simplest case $r=2, a\_1 = a\_2 = 1$, $i = 1$. By the formula in the Corollary at pag 407 we get $c\_q(D) = \sigma\_1 - \sigma\_0 = -\frac{1}{4}-\frac{1}{4} = -\frac{1}{2}$.


On the other hand by the formula in the end of pag 409 we get $c\_q(D)=-\frac{1}{4}$.


Does anyone know what is wrong here? Am I misunderstanding anything?



A question on young persons guide to canonical singularities
The last formula on page 409 clearly has a typo. There should be no $r$ in the denominator. If you look at the computation, (or even better, do it yourself!!), then you see that $r$ never appears in the denominator (they are always explicit numbers). In particular, this formula is based on the formula in the Proposition in (8.9) where it appears correctly as $-\frac{i(r-i)}2$. In fact, it appears correctly in the proof of the theorem with which you're having trouble (see the two lines starting with (II) just before (9.4) on page 410).


