Terminal Quotient Singularities in Dimensions Three and Four

We classify isolated terminal cyclic quotient singularities in dimension three, and isolated Gorenstein terminal cyclic quotient singularities in dimension four. In addition, we give a new proof of a combinatorial lemma of G. K. White using Bernoulli functions. Let A' be a smooth algebraic variety over C, and let ux be the canonical bundle of X. For each n > 0, if T(X, to®") ^ 0, there is a natural pluricanonical map PT( X, o)®")*. An algebraic variety is of general type if i>n is a birational map for n sufficiently large. For a variety of general type, the pluricanonical images n(X) are the most natural birational models of X to study. Canonical singularities are the singularities which may occur in the pluricanonical models of varieties of general type. In dimension 1, the pluricanonical models are smooth so there are no canonical singularities; in dimension 2 the canonical singularities coincide with the classical rational double points. One characterization of the rational double points is as quotient singularities: if G is any finite subgroup of Sl(2, C), then the quotient C2/G has a rational double point, and every rational double point is analytically isomorphic to such a quotient singularity. Reid and Shepherd-Barron [10], and independently Tai [14], have given a condition for quotient singularities to be canonical in arbitrary dimensions (although not all canonical singularities are quotient singularities in dimensions greater than two). Terminal singularities are a class of canonical singularities which play an important role in birational geometry (as evidenced by recent work of Mori [8], Reid [12], and Tsunoda [15]). In this note we study cyclic quotient singularities which are terminal. In dimension three we explicitly describe all isolated terminal cyclic quotient singularities, while in dimension four, we describe isolated terminal cyclic quotient singularities which are also Gorenstein. The description uses a combinatorial lemma due to G. K. White [17]; we have given a new proof of this lemma (Corollary 1.4 below) using Bernoulli functions. Received by the editors March 31, 1983. 1980 Mathematics Subject Classification. Primary 14B05; Secondary 10A40, 14L30, 32B30, 52A25. Key worás and phrases. Bernoulli functions, canonical singularity, Gorenstein ring, quotient singularity, terminal singularity. 1 National Science Foundation Postdoctoral Fellow. 2 Partially supported by National Science Foundation Grant MCS 82-01762. ©1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page 15 16 D R MORRISON AND GLENN STEVENS 1. Bernoulli functions. The material in this section is all quite standard (except for the last two corollaries); we have adapted the presentation in the first chapter of Kubert and Lang [7] to suit our purposes. If x is a rational number, we let (x) denote the rational number such that x = (x) modZ, and 0 < (*)< 1. Define , \ _ j (x)— | if x is not an integer, 1 [0 if x is an integer. Let N be a natural number, let C(N) = (l/N)Z/Z, and let G(N) = (Z//YZ). G(N) acts on C(N) by multiplication, and there is an embedding G(N) -> C(N) given by a -» a/N. Let x: G(N) -» C* be a character of conductor N and define B,,x= 2 x(«)B,(f). aeG(N) The following classical theorem is essentially due to Dirichlet; a nice proof can be found in [5, §2, Theorem 2]. Theorem 1.1. If x is an odd character (that is, x(~a) = ~x(a) for a^ a £ G(N), where N is the conductor of \), then Bx x=£ 0. Now fix a natural number N > 2, and denote C(N) and G(N) by C and G, respectively. Let V = C(G) be the group algebra of G generated (as a C-vector space) by elements aa for a G G. We define a function 5: C -» Kby S(jc) = 2 B,(ax)oa. Let If be the subspace of V generated by(5(x):x G C}, and let A = Ann( W) C V*. For each a G G, let Xa = a* + o*a G V*. Note that \a(S(x)) = Bx(ax) + Bx(-ax) = 0 for all x since B, is an odd function. Thus, Xa G A. Proposition 1.2. A is generated (as a C-vector space) by {\a: a G G). In particular, dimc(A) = dimc(W) — ip(N)/2, where <¡> is Euler's ^-function. Proof. Let x be an (arbitrary) odd character on G, M the conductor of x. and H = G(M). There is a well-defined map H -> C = (l/N)Z/Z given by a -* a/M. We define an element vvx G H7 by