Research on rationally connected varieties

为了研究高维代数簇X，我们利用Lefschetz定理，取X的一个超平面截面A，并试图通过A的超平面截面来寻找X的结构。这一次，我们研究了A的丛结构是否保持到X，接下来，爆破结构也是这样。作为推广，我们还研究了极值射线的保持性。作为应用，我们得到了如下结果：定理。让我们考虑光滑射影簇序列{X_n}，使得X_n S是X_(n+1)中的一个充分因子，这里n遍历每个正整数。设X_1有一个初等压缩f：X_1-&gt；Y，其中dim X_1-dim Y&gt；1和dim X_1&gt；2，则对每个n都有一个归纳扩张态射f_n：X_n-&gt；Y，其中f_{n-1}=i_(n-1)f_n，其中i_(n-1)：X_{n-1}-gt；X_{n}是自然嵌入。对于Y的一个非常一般的点y，f_n的一个光滑纤维是对足够大的n的一个加权完全交.上述定理表明，一个具有光滑射影簇序列{X_n}的簇具有“对称性”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".