Wild Quotient Singularities of Surfaces

Let (B,MB) denote a regular local ring of dimension two. Let H be a finite cyclic group acting on B, and let Z := Spec(B). Assume that the action of H on Spec(B) is free off the closed point, and thatMB is the only singular point of Z. Let f : X → Z be a resolution of the singularity, minimal with the property that the irreducible components of f(MB ) are smooth with normal crossings. Such a resolution exists when B is excellent ([1], [5], [17]). Let G denote the resolution graph of f : the vertices of G are the n irreducible components of f(MB ), and two vertices C and D are linked by (C ·D)X edges. The degree of a vertex C is then ∑ D 6=C(C · D)X , and a vertex of degree at least 3 is called a node. Let N denote the associated intersection matrix. When the order of H is not divisible by the residue characteristic p of B/MB, MB is called a tame cyclic quotient singularity. Otherwise,MB is a wild cyclic quotient singularity. Much is known about tame cyclic quotient singularities. Most importantly, their resolution graphs are Hirzerbruch-Jung strings, and for a given H , the set of such strings is finite. These facts are well-known for surfaces over C (see, e.g., [7], III.5, or [20], p. 207), but the general case does not seem to be fully stated and proved in the literature (see, however, [45], 6.4 and 6.8, and [14], section 2). The key property of tame cyclic actions that allows for a description of their resolution graphs is that such actions can be given in simple normal forms, producing explicit equations for the ring B . Wild cyclic actions are much more difficult to classify. For instance, for a given prime p, the set of non-singular matrices arising as intersection matrices of the resolution of Z/pZ-quotient singularities is infinite (8.5). Explicit equations for k[[x, y]] have only been obtained in a few cases, and only when |H| = p, such as in equicharacteristic 2 in [3], and in some cases where p = 3 in [33], 5.15. In this article, we first discuss three properties that a general intersection matrix is likely to possess in order to possibly be the intersection matrix associated with a cyclic quotient singularity. We show in certain rather general situations that the irreducible components of the resolution of a Z/pZ-quotient singularity are rational, and that the graph of the resolution is always a tree (Theorem 2.5). It would be of interest to know whether, given a prime p, there always exists, for any integer v, a resolution of a Z/pZ-quotient singularity whose graph has more than v nodes. We show also in certain rather general situations that the multiplicity of the local ring B is at most |H| (Theorem 2.3). This latter result imposes additional restrictions on the possible resolution graphs since the fundamental cycle Z of a singularity depends only on the intersection matrix, and its self-intersection |Z2| is bounded by the multiplicity of B ([46], 2.7). Finally, we show in certain rather general situations that |H| kills the finite component group ΦN := Z /N(Z) (Theorem 2.4). In particular, det(N) is a power of p when H = Z/pZ. It would be interesting to determine whether a non-singular intersection matrix