On \((-P\cdot P)\)-constant deformations of Gorenstein surface singularities (Q1884767)

This paper continues the study of deformations of Gorenstein surface singularities from the previous author's papers [Proc. Am. Math. Soc. 129, 2823--2831 (2001; Zbl 0983.14014), Int. J. Math. 12, 49--61 (2001; Zbl 1064.14002)]. Let \(T\) be a neighborhood of the origin of \({\mathbb C}\) and \(\pi:X\to T\) a small deformation of a normal Gorenstein surface singularity such that \(X_0=\pi^{-1}(0)\) is not log-canonical. If a topological invariant \(-P_t\cdot P_t\) of \(X_t=\pi^{-1}(t)\) is constant then after a finite base change, there exists a section \(\gamma:T\to X\) of \(\pi\) such that each \(\gamma(t)\) is a non-log-canonical singularity, and a simultaneous resolution \(f:M\to X\), which induces a locally trivial deformation of each maximal string of rational curves at an end of the exceptional set of \(M_0\to X_0\). Moreover, if the resolution graph of \((X_0,x_0)\) is star-shaped then each \(X_t\) has just one singular point and \(\pi\) is an equisingular deformation.