The canonical covers of normal two-dimensional rational triple points

Given Pinkham-Demazure’s description R=R(P1,D) of normal two- di-mensional rational triple point R, we can calculate systematically the canonical cover S=R(Y,˜D) by our theory of graded cyclic cove r[19] as in Theorem(1.5). If pg(S)=0 then R is aq uotient singularity and there are infinitely many such cases Theorem (2.3). However, in the case of pg(S)=1, we can show the finiteness of such R in Theorem (3.1). Further in Theorems (2.4) and (2.6), some special infinite series of rational triple points of A-types, we will see the finiteness of R when the invariants a(S)=a′/s or pg(S) are bounded from above. Our results of the present paper give a good evidence of affirmative direction for finiteness problem about weights of singularity which have been developed in recent studies [17,25].