Vanishing theorems, a theorem of Severi, and the equations defining projective varieties
Vanishing theorems, a theorem of Severi, and the equations defining projective varieties
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DOI：
10.1090/s0894-0347-1991-1092845-5
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发表时间：
1991-09
影响因子：
3.9
通讯作者：
A. Bertram;L. Ein;R. Lazarsfeld
中科院分区：
文献类型：
--
作者：
A. Bertram;L. Ein;R. Lazarsfeld
1903年Severi [S]证明了：如果X，X'cP”是光滑曲面，其并集是二次r ~ 2超曲面的完全交，则X，X' cP”是二次r ~ 2超曲面的完全交。，dr-2，则k > Z di r的超曲面在X上切出一个完备的线性级数（参见[SR，XIII.9.8]）。本文的目的是首先表明，基本参数使用科代拉消失定理导致一个简单的变种塞韦里的声明（定理7以下），它在几个方向上扩展。更重要的是，我们希望说服读者，这一结果有一个令人惊讶的数量的应用程序的问题，涉及方程定义射影品种。考虑开始与光滑复射影簇X C Pr的维数n和余维数e = r n。在这种情况下，我们的定理断言了4月命题1中X的理想层Yx的ath幂的适当扭曲的高阶上同调的消失。假设X在P”中被d1次的超曲面切出？d2> > d_。则对于i > 1，如果k > d1 + d2+ + de r，则H（P 'r，（k））= 0。注意，这里只有第一个e = codim（X，Pr）定义方程的次数起作用。当n = 2和a = 1时，这是塞维里的结论，
In 1903 Severi [S] proved that if X, X' c P" are smooth surfaces whose union is the complete intersection of r 2 hypersurfaces of degrees di ... , dr-2, then hypersurfaces of degree k > Z di r cut out a complete linear series on X (cf. [SR, XIII.9.8]). The purpose of this paper is first of all to show that elementary arguments using the Kodaira vanishing theorem lead to a simple variant of Severi's statement (Theorem 7 below) which extends it in serval directions. More importantly, we hope to convince the reader that this result has a surprising number of applications to questions involving the equations defining projective varieties. Consider to begin with a smooth complex projective variety X C Pr of dimension n and codimension e = r n. In this setting, our theorem asserts the vanishing of the higher cohomology of suitable twists of the ath power of the ideal sheaf .Yx of X in Apr. Proposition 1. Assume that X is cut out scheme-theoretically in P" by hypersurfaces of degrees d1 ? d2> > d_. Then H (P'r,(k)) = 0 for i > 1 provided k > ad, + d2+ + de r. Note that only the degrees of the first e = codim(X, Pr) defining equations come into play here. When n = 2 and a = 1 this is a consequence of Severi's