Weighted homogeneous surface singularities admitting rational homology disk smoothings

Abstract

We study rational homology disk smoothings through canonical-cover representations and specialization of reflexive canonical powers, and recover the known dual graph and analytic classification. The geometric-genus representation of the canonical covering group, together with its complex conjugate, is a direct summand of the reduced regular representation. Specialization forces the canonical Riemann--Roch corrections to vanish; explicit formulas give equations for the Seifert invariants. For three arms and central weight two, these equations, the character restrictions, and the numerical identities determine the Seifert data without assuming the existence of a smoothing. Projective compactifications treat the remaining cases, and elliptic pencils explain the four-arm attachment points and the parameter independence of their cross ratios. We construct strong $\mathbb{Q}$-Gorenstein smoothings for all listed analytic structures by deforming canonical-cover equations and lifting generators of the full first syzygy module. The numerical Milnor formula gives rational homology disk fibers.