Unexpected Stein fillings, rational surface singularities and plane curve arrangements

We compare Stein fillings and Milnor fibers for rational surface singularities with reduced fundamental cycle. Deformation theory this class of was studied by de Jong-van Straten in [dJvS98]; they associated a germ singular plane curve to each singularity described via deformations curve. consider links singularities, equipped their canonical contact structures, develop symplectic analog Straten's construction. Using planar open books Lefschetz fibrations, we describe all the certain arrangements disks, related homotopy singularity. As consequence, show that many admit are not strongly diffeomorphic any fibers. This contrasts previously known cases, such as simple quotient where give rise fillings. On other hand, if cycle, self-intersection exceptional is at most -5 minimal resolution, then link has unique filling (given fiber).