A Geometric Characterization of Normal Two-Dimensional Singularities of Multiplicity Two with $p_a ≤ 1$

Abstract

IntroductionA normal two-dimensional singularity (V, p) is a germ of normal two-dimensional complex analytic space V with a reference point p. Let (V, p)< -(V, A} be a resolution of (V, p) with exceptional set A. The geometric genus of a normal two-dimensional singularity (V, p) is the integer p g (V, p} defined byThe arithmetic genus of a normal two-dimensional singularity (V, />) is the integer p a (V, p} defined by Pa(V, £)=sup p a (D). D>0Here, the integer p a (D) is the virtual genus of the divisor D on V. These two integers are independent of the choice of the resolutions (see [11] [18] The condition p g = Q is equivalent to the condition p a = 0 (Artin [2] [3]). For the singularity of multiplicity two, it satisfies the condition p g = G if and only if it can be resolved by a succession of blowing-ups at point (i. e., it is an absolutely isolated singularity) (Satz 1. of Brieskorn [4]).The goal in this paper is a characterization of the normal twoCommunicated by S. Nakano, November 24, 1981. Revised November 28, 1982 l the blowing up of V l at a point in the singular locus of y x , T 2 ; F 2 > F 2 the normalization of V 2 , and so on.Moreover, this process ends in finite steps. The result of this paper is the following. The proof of the theorem in this paper is not based on the classification as above, but is based on the explicit computations in the reduction cf the singularity to the absolutely isolated singularities NORMAL TWO-DIMENSIONAL SINGULARITIES 3 (in the proof of Lemma 5). Theorem. Let (V, p) be a normal two-dimensional singularity of multiplicity two. The condition p a^l is satisfied if and only if the normalization T { is trivial or is obtained by a blowing up of VIn § 1, the construction of a sequence of modifications, which plays an important role in this paper, is given. In §2, the proof of the theorem is given. In §3, the computation which is essential in this paper is done. In §4, three remarks are given.The author gives his thanks to Le Dung Trang, A. Fujiki, Kimio Watanabe, I. Naruki, and K. Saito for many suggestions and encouragements. § 1. A Canonical Resolution for the Normal Two-Dimensional Singularity of Multiplicity Two (1. 1) In this section, for a normal two-dimensional singularity of multiplicity two, a resolution by a sequence of modifications is constructed, which plays an important role in this paper. Some preliminary remarks on the resolution are given. This method is rather standard (e. g., Kirby [9], Horikawa [8], Laufer [13]), and is useful for the study to see how the singularity becomes the absolutely isolated singularities, step by step.(1.2) Let (F, p} be a normal two-dimensional singularity of multiplicity two represented as {(x, y, z)^U\z 2 -g(x, 3;) =0} (cf.Introduction). After the blowing up of V at p, the following commutative diagram follows.U*^-U, \j w "I Here, the analytic space H is the hyperplane in C 3 defined by {z^O}, the map TT is the restriction to V of the projection from C 3 to H' (x, y, z) > O, y, 0) . The map r x is the blowing up of U at p, the a...

A $p_g$-Formula and Elliptic Singularities

“…In the case of mult p F=2, the ^-formula for the canonical resolution stated in Lemma 2 of [43] can be also induced from our formula (2.7) (cf. the argument in (4.8)).…”

“…See [4], [8], [26], [28], [30], [31], [43], [46], [48], [52] for the basic facts on two-dimensional singularities and those numerical invariants.…”

“…This is the method to compute the Milnor number (e.g., by the method [10]) and to relate it to p q by Laufer's formula [27]. § 3 e Maximal Ideal Cycles [43]). In this section we shall study the virtual arithmetic genus p a (Y lj/ ).…”

“…where \l/ t is the blowing-up of U i -l with smooth center F t contained in the singular locus Sing(l^-_ l ) of Vi-i and V t the strict transform of V i , 1 §2 of [43] or [18] for the existence of such a divisor). We shall consider in the neighborhood of the inverse image of {p} in each stage.…”

“…After his definition, a normal two-dimensional singularity (K, p) is called elliptic if the condition p a (V 9 p)=\ holds. In this decade, a great deal of work has been done on this singularity (e.g., [28], [34], [43], [48], [49], [50], [51],..., etc). Among others, H. B. Laufer has given the criterion for the absolute isolatedness (cf.…”