General hyperplane sections of nonsingular flops in dimension 3

Let X be a 3-dimensional complex manifold, and f : X → Y a proper bimeromorphic morphism to a normal complex space which contracts an irreducible curve C ⊂ X to a singular point Q ∈ Y while inducing an isomorphism X \C ≃ Y \ {Q}. We assume that the intersection number with the canonical divisor (KX ·C) is zero. In this case, it is known that the singularity of Y is Gorenstein terminal, and there exists a flop f : X → Y ([R]), which we call a nonsingular flop because X is nonsingular. In order to investigate f analytically, we replace Y by its germ at Q, and consider a general hyperplane section H of Y through Q. Then H has only a rational double point, its pull-back L ⊂ X by f is normal, and the induced morphism fH : L → H factors the minimal resolution g : M → H ([R]). The dual graph Γ of the exceptional curves of g is a Dynkin diagram of type An, Dn or En. Let F = ∑n k=1 mkCk be the fundamental cycle for g on M . The natural morphism h : M → L is obtained by contracting all the exceptional curves of g except the strict transform Ck0 of C. Kollár defined an invariant of f called the length as the length of the scheme theoretic fiber f(Q) at the generic point of C. It coincides with the multiplicity mk0 of the fundamental cycle at Ck0 . Katz and Morrison proved the following theorem ([KM, Main Theorem]). The purpose of this paper is to give its simple geometric proof.