With the notation of [Reference YasudaYas14], the following is the statement of [Reference YasudaYas14, Corollary 6.21].
Theorem 1. Suppose that V has no reflection and that there exists a crepant resolution
$f\colon Y\to X$
. Then the following hold:
-
(i)
$\mathrm{M}_{\mathrm{st}}(X)=[Y]$
; -
(ii)
$D_{V}=p$
; and
-
(ii)
$e_{\mathrm{top}}(Y)=p$
.
The original proof of the second and third assertions contains an error. In that proof, we use the formula
\begin{align}e_{\mathrm{st}}(X)=1+\frac{p-1}{D_{V}-p+1}\end{align}
from Corollary 6.12 of op. cit., and use the fact that this value is an integer, since
$e_{\mathrm{st}}(X)=e_{\mathrm{top}}(Y)$
. The author then argued, incorrectly, that the denominator in the formula must be 1, from which the second assertion and then the third assertion would follow. However, the right-hand side of (1) is an integer as long as the denominator is a divisor of
$p-1$
, so the proof is wrong. We give a correct proof as follows.
Proof. The quotient variety X is factorial [Reference Campbell and WehlauCW11, Theorem 3.8.1]; in particular,
$\mathbb{Q}$
-factorial. Any resolution of X has exceptional locus of pure codimension one [Reference KollárKol96, Chapter VI, Theorem 1.5]. From [Reference KollárKol13, Corollary 2.12] and the existence of a crepant resolution, we see that X is canonical but not terminal. From [Reference YasudaYas19, Corollary 1.4], we have
$D_{V}=p$
. From (1), we conclude that
$e_{\mathrm{top}}(Y)=e_{\mathrm{st}}(X)=p$
. Note that the above argument is not circular: although the work [Reference YasudaYas19] relies on several results from [Reference YasudaYas14], it does not depend on Corollary 6.21.
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