Research on rationally connected varieties

对于高维代数变量X的研究，我们利用Lefschetz定理取X的一个超平面截面a，用a的超平面截面a求出X的结构，这一次我们研究了a的束结构是否保留到X，接下来我们研究了膨胀的结构是否保留。在此基础上，进一步研究了极值射线的保存问题。作为应用，我们得到如下定理。让我们考虑一个光滑射影变体的序列{X_n}，使得X_n是X_{n+1}中每个n的一个样本约数。这里n遍历每个正整数。假设X_1有一个初等收缩函数f: X_1 -> Y具有微弱的X_1 -微弱的Y > 1和微弱的X_1 > 2。然后对于每一个n存在一个归纳扩展态射f_n: X_n -> Y，其中f_{n-1}=i_{n-1}f_n，其中i_{n-1}: X_{n-1} -> X_{n}是一个自然嵌入。对于y的一个非常一般的点y， f_n的光滑纤维对于足够大的n是一个加权的完全交。上述定理表明，具有光滑射光变分序列{X_n}的变分具有“对称”性质的结构。

For the study of higher dimensional algebraic varieity X we take a hyperplane section A of X for the use of Lefschetz Theorem and try to find the structure of X by the one of A.This time we studied whether the bundle structure of A is preserved to X and next the structure of blowing-up is also so. Moreover generalizing the method,we investigate the preservation of the extremal ray.As applications we get the following : Theorem. Let us consider a sequence {X_n} of smooth projective varieties so that X_n s an ample divisor in X_{n+1} for each n. Here n runs over each positive integer. Assume X_1 has an elementary contraction f : X_1 -> Y with dim X_1 - dim Y > 1 and dim X_1 > 2.Then for each n there is an inductively extended morphism f_n : X_n -> Y with f_{n-1}=i_{n-1}f_n where i_{n-1} : X_{n-1} -> X_{n} is a natural embedding. For a very general point y of Y a smooth fiber of f_n is a weighted complete intersection for large enough n.The above theorem says that a variety enjoying a sequence {X_n} of smooth projective varieties has the structure of property "symmetry".