Comparison of fundamental cycles and maximal ideal cycles for normal surface double points—due to Laufer decomposition

For a resolution space $(\tilde{X}, E)$ of a normal complex surface singularity $(X, o)$, the fundamental cycle $Z_{E}$ and maximal ideal cycle $M_{E}$ are important geometric objects associated to $(X, o)$, which satisfy $M_{E} \geqq Z_{E}$. In 1966, M. Artin proved that $M_{E} = Z_{E}$ for all resolutions of all rational singularities. However, for non-rational singularities, it is a delicate problem whether $M_{E} = Z_{E}$ or not. Any normal surface double point (i.e., multiplicity two) is a hypersurface singularity defined by $z^{2} = f(x, y)$. For such singularities, we prove that $M_{E} > Z_{E}$ holds on the minimal resolution if and only if $f$ has a canonical decomposition $f = f_{[L]} f_{[c]} f_{[o]}$ in $\mathbb{C}\{x, y\}$ called “Laufer decomposition”. Moreover, we give a numerical procedure to determine whether $M_{E} = Z_{E}$ or not on the minimal resolution from the embedded topology of the branch curve singularity $(\{f = 0\}, o)$.