On the estimate of the arithmetic genus for normal two-dimensional singularities on double coverings

Abstract

In this paper, we deal with normal two-dimensional singularities with multiplicity two. We call such a singularity double point. The purpose of this paper is to give an estimate of the arithmetic genus for double points in terms of Horikawa's canonical resolution and a $p_{a}$-formula for some class of double points. It is known that, by using the data obtained from the canonical resolution, the geometric genus for double points is formulated, and rational double points and elliptic double points are characterized. We give a characterization of double points with the arithmetic genus two.