arXiv:alg-geom/9505035v1 31 May 1995 SEMISTABLE 3-FOLD FLIPS Alessio Corti December 3, 1994 1 Introduction. In this short note, I will give a proof of the existence of semistable 3-fold flips, which does not use the classification of log terminal (i.e., quotient) surface singularities. This permits us to avoid a calculation, which is a sort of logical bottleneck in all the existing approaches to semistable 3-fold flips ([Ka1 pg. 158–159], [Sh pg. 386–389], [Ka3 pg. 483–486]), however different they may look. The message is that Shokurov’s main reduction step, as refined in [FA, Ch. 18], can be used profitably in the semistable case also. My main motivation was to develop an approach to semistable flips that would have some fighting chances in dimension 4, and the present paper is a first (small) step in that direction. We always work over an algebraically closed field of characteristic zero. 1.1 Definition. Let X be a normal projective variety, B ⊂ X a Q-Weil divisor such that K X + B is log terminal (this notion is recalled in 2.2). Let R ⊂ NE(X ) be an extremal ray with (K + B) · R< 0, ϕ R : X → U the contraction of R. ϕ R is said to be a flipping contraction if the ϕ R -exceptional locus has codimension ≥ 2. A flip of ϕ R is by definition a variety X + , together with a morphism ϕ + : X + → U , such that K + + B + is Q-Cartier and ϕ + -ample. It is easy to see that the flip is unique if it exists. It is not so easy, but true, that K + + B + is log terminal (see 2.4). The flip conjecture asserts that flips exist and that there is no infinite sequence of them. An important special class of flips is that of semistable flips. These are the flips that appear in the minimal model program for a semistable family. In short, one is given the additional structure of a projective morphism f : X → Δ, where Δ = Spec O is the spectrum of a discrete valuation ring O, with central and generic points 0,η ∈ Δ. We denote X 0 , X η the fibers over 0, η. All extremal rays, contractions, flips, are compatible with this structure. The starting point is a semistable family in the sense of Mumford, but the minimal model program will soon introduce singularities, which we call semistable terminal Typeset by A M S-T E X 1