On (-P P)-constant deformations of Gorenstein surface singularities

Let π : X →T be a small deformation of a normal Gorenstein surface singularity X 0 =π-1(0) over the complex number field ℂ.Suppose that T is a neighborhood of the origin of ℂ and thatX 0 is not log-canonical.We show that if a topological invariant-P t ⋅ P t of X t = π-1(t) is constant, then, after a suitable finite base change,π admits a simultaneous resolution f : M → X which induces a locally trivial deformation of each maximal string of rational curves at an end of the exceptional set of M 0→ X 0;in particular, if X 0has a star-shaped resolution graph, then πadmits a weak simultaneous resolution (in other words, π isan equisingular deformation).