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

Abstract

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 ≧ 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 𝑧² = 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 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).