On Gorenstein surface singularities with fundamental genus \(p_ f\geq 2\) which satisfy some minimality conditions. (English) Zbl 0848.14017

Tomaru, Tadashi

Pac. J. Math. 170, No. 1, 271-295 (1995).

In this paper we study normal surface singularities whose fundamental genus (:= the arithmetic genus \(p_a (Z)\) of the fundamental cycle \(Z)\) is equal or greater than 2. For those singularities, we define some minimality conditions which are similar to Laufer’s conditions for elliptic singularities, and we study the relation between them under the condition to be Gorenstein. Especially, we prove that
“If \(\pi : (\widetilde X,A) \to (X,x)\) is a resolution of a Gorenstein surface singularity with \(p_g (X,x) \geq 1\) and \(K = Z + E\), then \(p_g (X,x) = p_a (Z) + 1\), where \(K\) and \(E\) are the canonical cycle and minimal cycle, respectively”.
Further we define some sequence of such singularities, which is analogous to elliptic sequence in the sense of Yau. In the case of hypersurface singularities of Brieskorn type, we study some properties of the sequences.

Reviewer: T.Tomaru (Maebashi)

MSC:

14J17	Singularities of surfaces or higher-dimensional varieties
14J70	Hypersurfaces and algebraic geometry
14E15	Global theory and resolution of singularities (algebro-geometric aspects)

Keywords:

normal surface singularities; fundamental genus; resolution; Gorenstein surface; hypersurface singularities