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)]).