General hyperplane sections of nonsingular flops in dimension 3
General hyperplane sections of nonsingular flops in dimension 3
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维度 3 中非奇异触发器的一般超平面截面
DOI：
10.4310/mrl.1994.v1.n1.a6
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发表时间：
1993
影响因子：
1
通讯作者：
Y. Kawamata
中科院分区：
文献类型：
--
作者：
Y. Kawamata
设\(X\)是一个三维复流形，\(f:X\to Y\)是到一个正规复空间的一个恰当双有理态射，它将\(X\)中的一条不可约曲线\(C\subset X\)收缩到\(Y\)中的一个奇点\(Q\in Y\)，同时诱导出一个同构\(X\setminus C\simeq Y\setminus\{Q\}\)。我们假设与典范除子的交数\((K_X\cdot C)\)为零。在这种情况下，已知\(Y\)的奇点是戈伦斯坦终端奇点，并且存在一个翻转\(f:X\to Y\)（[R]），由于\(X\)是非奇异的，我们称其为非奇异翻转。为了对\(f\)进行解析研究，我们用\(Y\)在\(Q\)处的芽来代替\(Y\)，并考虑\(Y\)通过\(Q\)的一个一般超平面截面\(H\)。那么\(H\)只有一个有理双点，它通过\(f\)的拉回\(L\subset X\)是正规的，并且诱导的态射\(f_H:L\to H\)分解了极小分解\(g:M\to H\)（[R]）。\(g\)的例外曲线的对偶图\(\Gamma\)是\(A_n\)、\(D_n\)或\(E_n\)型的 Dynkin图。设\(F = \sum_{k = 1}^{n}m_kC_k\)是\(M\)上\(g\)的基本圈。自然态射\(h:M\to L\)是通过收缩\(g\)的所有例外曲线（除了\(C\)的严格变换\(C_{k_0}\)）得到的。科拉尔定义了\(f\)的一个不变量，称为长度，它是\(C\)的一般点处的概型理论纤维\(f(Q)\)的长度。它与基本圈在\(C_{k_0}\)处的重数\(m_{k_0}\)一致。卡茨和莫里森证明了以下定理（[KM，主定理]）。本文的目的是给出其简单的几何证明。
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.