The pluri-genera of surface singularities
\textit{K. Watanabe} [Math. Ann. 250, 65-94 (1980; Zbl 0414.32005)] introduced the pluri-genera \(\{ \delta_m (X,x)\}_{m \in \mathbb{N}}\) of a normal isolated singularity \((X,x)\). They can be computed from a good resolution \(f:M\to X\) (i.e. a resolution whose exceptional locus \(A\) is a divisor with normal crossings) as follows: \[ \delta_m (X,x)= \dim_{\mathbb{C}} H^0 ({\mathcal O}_U(mK)) / H^0 ({\mathcal O}_M (mK+(m-1)A)) \] where \(K\) is a canonical divisor on \(M\) and \(U=X \setminus \{x\} \cong M \setminus A\). In particular, \(\delta_1 (X,x) = p_g (X,x)\). The author studies the pluri-genera of normal surface singularities over \(\mathbb{C}\). For rational surface singularities whose dual graph for the minimal resolution is a star-shaped graph, he describes the pluri-genera in terms of some divisors on the central exceptional curve. Using this, he proves that, if for a normal surface singularity \((X,x)\) over \(\mathbb{C}\), \(\delta_m(X,x)=0\) for \(m=4,6\) then \((X,x)\) is a quotient singularity. Since K. Watanabe proved that \((X,x)\) is a quotient singularity if and only if \(\delta_m(X,x)=0\) for all \(m \in \mathbb{N}\), the preceding result characterizes the quotient singularities. Purely elliptic singularities (i.e. those for which \(\delta_m(X,x)=1\) for all \(m \in \mathbb{N}\)) are also characterized in the paper. It is proved that a normal surface singularity \((X,x)\) over \(\mathbb{C}\) is purely elliptic if and only if \(\delta_m(X,x)=1\) for \(m=1,4,6\). Applying the result of S. Ishii which assures that \((X,x)\) is purely elliptic if and only if it is a cusp or a simple elliptic singularity, the result in the paper gives a criterion to characterize these singularities. S. Ishii had also proved that \((X,x)\) is log-canonical if and only if \(\delta_m(X,x) \leq 1\) for all \(m \in \mathbb{N}\). In the paper, the author proves that if \(\delta_{14}(X,x)=0\) then \((X,x)\) is log-canonical.
scientific article; zbMATH DE number 1185535
scientific article