Topological classification of simplest Gorenstein non-complete intersection singularities of dimension 2 (Q906889)

Let \(p\in V\) be a normal complex surface singularity and \(\Gamma\) the resolution graph, namely, the weighted dual graph of the exceptional set on the minimal good resolution \(M\). Let \(Z\) denote the fundamental cycle on \(M\). \textit{M. Artin} [Am. J. Math. 88, 129--136 (1966; Zbl 0142.18602)] proved that rational surface singularities are characterized by their resolution graph and that a rational surface singularity \((V,p)\) is a hypersurface singularity if and only if \(-Z^2=2\), and classified the resolution graphs of the rational hypersurface singularities. \textit{H. B. Laufer} [Am. J. Math. 99, 1257--1295 (1977; Zbl 0384.32003)] introduced the minimally elliptic singularities, which are characterized by their resolution graph, and proved that a minimally elliptic singularity \((V,p)\) is a hypersurface (resp. complete intersection) singularity if and only if \(-Z^2\leq 3\) (resp. \(-Z^2\leq 4\)), and classified the resolution graphs of all minimally elliptic hypersurface singularities. On the other hand, \textit{H. B. Laufer} [Rice Univ. Studies 59, No. 1, 53--96 (1973; Zbl 0281.32009)] also proved that, if \(\Gamma\) does not correspond to rational or minimally elliptic singularities, then the general surface singularities with resolution graph \(\Gamma\) are not Gorenstein. Note that a complete intersection singularity is Gorenstein. In the paper under review, the authors classify the resolution graphs of ``simplest Gorenstein non-complete intersection surface singularities'', that is, the minimally elliptic singularities with \(-Z^2=5\). The methods are the same as that in [\textit{F. Chung} et al., Trans. Am. Math. Soc. 361, No. 7, 3535--3596 (2009; Zbl 1171.32016)] in which the resolution graphs of the minimally elliptic singularities with \(-Z^2=4\) are classified.