Discrepancies and multiplicity of rational singularity

Question

Let $(X,x)$ be a rational normal surface singularity having multiplicity $m$ (for example $(-Z)^{2}$, where $Z$ is the fundamental cycle). Suppose its discrepancies are all $\ge -1+\frac{1}{k}$ for a $k \in \mathbb{Z}_{\ge 1}$, i.e. the singularity is $\frac{1}{k}$-log canonical.

Is there some nice connection between $m$ and $k$?

Answer 1

Easily to obtain the results: $mld(X) \le \frac{2}{m}$, where $mld(x) = \frac{1}{k}$, in toric case, you can check this by the steps of Hirzebruch-Jung Continued Fractions in the resolution of toric surface.