Remarks on the Geometric Genus of Hypersurface Isolated Singularity

The theory of toric varieties is a beautiful, powerful subject which is finding more and more uses. Unfortunately its complexity (i.e., the dual of the dual) makes it rather difficult to use. This short paper consists of some instructive elementary examples and applications collected as we were learning the subject. After reading this, the reader could go on to a detailed introduction such as [Oda], and then tackle Munford's paper. In this paper, we focus on the resolution of hypersurface isolated singularity by torus embbedings. This resolution can easily be constructed in the (special) cases when the defining equation is non-degenerate with respect to its Newton boundary. We then give an application of the resolution above. The geometric genus of a hypersurface isolated singularity is easily computed via resolution. The material of this paper was developed in discuss with K. Watanabe. We recall a few preliminaries related to the concept of the geometric genus of normal isolated singularities in §1 and torus embedding in §2. In §3, we show the figure of the resolution of hypersurface isolated singularities