A geometric characterization of a simple $K3$-singularity

Abstract

A simple ^-singularity is a three-dimensional normal isolated singularity with a certain condition on the mixed Hodge structure on a good resolution. We prove here that a three-dimensional normal isolated singularity is a simple ^-singularity if and only if the exceptional divisor of a β-factorial terminal modification is an irreducible normal Λ3-surface. A simple ^-singularity is defined in terms of the Hodge structure as a threedimensional analogue of a simple elliptic singularity. It is well known that a simple elliptic singularity is characterized by the geometric structure of the minimal resolution (cf. [S], [II] and [Wl]). The aim of this paper is to prove that a simple ^-singularity is also characterized by the geometric structure of a (J-factorial terminal modification which is a three-dimensional analogue of the minimal resolution (cf. [M]). This characterization should help investigations of a simple A3-singularity which are being carried out from various viewpoints (cf. [T], [W2], [W3] and [Y]). The authors would like to thank Professors M. Tomari and K-i. Watanabe for their helpful advices during the preparation of this paper. Let/: X-+X be a good resolution of a normal isolated singularity (X, x), where a resolution is called a good resolution if E=f~ 1(x)τed is a divisor with normal crossings. We decompose £into irreducible components Ei{i=\,2,..., s). If (A, x) is a Gorenstein singularity, then we have a presentation of canonical divisors