Two aspects of log terminal surface singularities
Log canonical singularities of a surface \(X\) have an index, i.e. a smallest positive integer \(I\) such that \(I\cdot K_X\) is Cartier. If \(\sigma:\widetilde{X}\to X\) is a good resolution of \(X\), \(E_i\) being the exceptional curves, \(I\) is obtained from the codiscrepancy \(\Delta\), where \(\sigma^*K_X= K_{\widetilde{X}}+ \Delta\) and \(\Delta=\sum \delta_iE_i\) and \(I=\min\{n\in \mathbb{N}_+\mid \forall i:n\cdot\delta_i\in \mathbb{Z}\}\). This paper studies log canonical and log terminal singularities, i.e. such singularities with \(\delta_i\leq 1\) for all \(i\) (\(\delta_i<1\), respectively), for a fixed number \(I\). There is a classification given in theorem A: The set of log terminal singularities consists of finitely many series (the number of series is explicitly given for each \(I\)) and in each series the singularities are indexed by natural numbers and ``behave very regular''. Essential for this investigation is a new numerical characterization of cyclic quotient singularities: The Hirzebruch-Jung continued fraction and the description of Riemenschneider, respectively, are compared with a triplet \((I,m,\alpha)\) involving the index \(I\), where \(n=I\cdot m\), \(q=m\cdot\alpha-1\) if the singularity is of type \((n,q)\). A series of cyclic quotient singularities is obtained simply in the form \((I,m,\alpha), (I,m+I,\alpha), (I,m+2I,\alpha),\dots\) -- The theorem provides an algorithm to obtain all log canonical singularities of given index \(I\). The author points out that his recent implementation on a computer proved useful to find log surfaces of prescribed index and Kodaira dimension 0. In theorem B, the weights \(b_i:= E_i^2\) of the singularity are considered. They are shown to measure ``how singular'' is the singularity for the case of log terminal singularities, whereas this is not the case in general. -- As an application, a new bound for the maximal number of weights \(\neq 2\) of a log terminal singularity of index \(I\) is given. Further, ascending and descending chain conditions for maximum and minimum, respectively of the \(\delta_i\) are considered. The author points out a partial overlap with a paper by \textit{A. Alexeev} [Duke Math. J. 69, No. 3, 527-545 (1993; Zbl 0791.14006)].
scientific article; zbMATH DE number 706663
scientific article