On Four - Dimensional Terminal Quotient Singularities

We report on an investigation of four-dimensional terminal cyclic quotient singularities which are not Gorenstein. (For simplicity, we focus on quotients by cyclic groups of prime order.) An enumeration, using a computer, of all such singularities for primes < 1600 led us to conjecture a structure theorem for these singularities (which is rather more complicated than the known structure theorem in dimension three). We discuss this conjecture and our evidence for it; we also discuss properties of the anticanonical and antibicanonical linear systems of these singularities. Introduction. The recent successes in understanding the birational geometry of algebraic varieties of dimension greater than two have focused attention on a class of singularities (called terminal singularities) which appear on the birational models which the theory selects. In dimension three, the structure of these terminal singularities is known in some detail: The terminal quotient singularities were classified, in what is now called the "terminal lemma", by several groups of people working independently (cf. [1], [2], [5], [8]), and all other three-dimensional terminal singularities were subsequently classified by work of the first author [6] and of Kollár and Shepherd-Barron [4]. Both of these classifications were based on key results of Reid [10], [11] who reduced the problem to an analysis of quotients of smooth points and double points by cyclic group actions. (A detailed account of the classification can be found in [12].) A consequence of this classification, apparently first observed by Reid [12], is that for any three-dimensional terminal singularity T, the general element of the anticanonical linear system | KT\ has only canonical singularities. This in turn implies that if we form the double cover of T branched on the general antibicanonical divisor D E [ — 2Kt\, that double cover will also have only canonical singularities. At first glance, this second property appears to have no advantages over the first, but looking at things in these terms proved decisive in a slightly more global context: Kawamata [3] showed that if T is an "extremal neighborhood" of a rational curve C on a threefold, then the existence of such a divisor D globally on T is sufficient to conclude the existence of a certain kind of birational modification (a "directed flip") centered on C. The first author [7] showed that such divisors always exist, completing the proof of the Minimal Model Theorem for threefolds. (See [7] for more details.) Received August 26, 1987; revised December 17, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 14B05; Secondary 14-04, 14J10, 14L32. •Research partially supported by the National Science Foundation. "Current addresses. S. M.: Department of Mathematics, Faculty of Science, Nagoya University, Nagoya 464, Japan; D. R. M.: Department of Mathematics, Duke University, Durham, North Carolina 27706; I. M.: Department of Mathematics, Fordham University, Bronx, New York 10458. ©1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page