Certain normal surface singularities of general type (Q1692606)

Let \((V,o)\) be a complex normal surface singularity, \(\pi: X\to V\) a resolution, \(p_a(V,o)\) its arithmetic and \(p_g(V,o)\) its geometric genus, \(Z_K\) the canonical cycle on \(\pi^{-1}(0)\). An unpublished result of Koyama states: If \(p_f(V,o)>1\), then \(-Z_K^2\geq 8p_a(V,o)-8\) and equality holds iff there exists a curve \(D\) on \(X\) such that \(Z_K=2D\) and \(p_a(D)=p_a(V,o)\) -- a proof of this result is given in Prop.\ 1.6 of the paper under review. A normal surface singularity \((V,o)\) with \(p_f(V,o)>1\) is called an \textit{even singularity} if there exists a resolution \(\pi: X\to V\) and a curve \(D\) on \(X\) such that \(Z_K=2D\). An even singularity is numerically Gorenstein. In this paper the author studies even singularities which have fundamental genus \(2\). For the rest of this review, let \((V,o)\) be an even singularity of genus \(2\), \(\pi: X\to V\) its minimal resolution, and let \(Z\) and \(Z_K\) be the fundamental cycle and the canonical cycle on \(X\), respectively. The singularity \((V,o)\) is a said to be a maximally even singularity if the quality \(p(V,o)=3p_a(V,o)-3\) holds. The minimal model [\textit{K.\ Konno}, J. Math. Soc. Japan 62, 467--486 (2010; Zbl. 1193.14047)] of \(Z\) is called the minimal even cycle. In section \(2\) the author classifies the weighted dual graph of \(Z\). A summary of the results -- in case \(p_a(V,o)=2\) -- is collected in Prop.\ 2.7: Minimally even cycles of fundamental genus \(2\) fall into five classes. The results in the case \(p_a(V,o)>2\) are collected in Prop.\ 2.8. In section 3 he establishes -- in Th.\ 3.2 -- a formula computing \(p_g(V,o)\) modeled on \textit{T. Okuma}'s formula [Math. Z. 249, 31--62 (2005; Zbl. 1091.32010)]. In section 4 the author studies the maximal ideal cycles of \((V,o)\); the main result is Th.\ 4.1 which states that even Gorenstein singularities of fundamental class 2 fall into \(4\) types. Finally, in section \(5\), Th.\ 5.2, the embedding dimension of a maximal singularity \((V,o)\) is computed. Reviewer's remark: In the proof of Lemma 2.4, (c), the author overlooked one more possibility [ibid. 24, No. 2, 293--294 (2017; Zbl 1461.14048)].