Flips arising as quotients of hypersurfaces

Abstract

Flips occur in the theory of minimal models of algebraic varieties. For an introduction and references see [1, lecture no. 5]. For varieties X− and X+, I denote the canonical class by K− and respectively. A flip is a diagram X−→X←X+ of normal complex quasiprojective 3-folds satisfying the conditions:1. both morphisms are birational and projective, contracting only finitely many curves C±⊂X± to an isolated singular point P∈X;2. the divisors −K− and K+ are relatively ample, that is, −K−Γ>0 for any curve Γ contracted by the morphism X−→X and similarly for K+;3. the two varieties X− and X+ have only terminal singularities.A diagram satisfying condition 1 is called a flip diagram. It is said to be directed by the canonical class if it also satisfies condition 2. Notice that condition 3 is overstated since under all the other conditions X+ will automatically have terminal singularities (see [4, 5-1-11(2)]).