Elementary transformations in the theory of algebraic vector bundles
Elementary transformations in the theory of algebraic vector bundles
复制标题
代数向量丛理论中的初等变换
DOI：
10.1007/bfb0071286
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发表时间：
1982
影响因子：
0.5
通讯作者：
M. Maruyama
中科院分区：
文献类型：
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作者：
M. Maruyama
导论.大约十年前，我提出了以下问题：“在高维射影簇上构造许多向量丛”。在考虑这个问题时，我发现了代数向量丛上的一种运算，一种初等变换。利用这种运算，在每个射影非奇异簇上构造了许多向量丛。另一方面，它最近出现的转换是一个强大的工具，在各个方向的理论代数向量丛。让我举一些例子：（1）这在变形理论和曲线上向量丛的模空间的去奇异化中发挥了关键作用（Narasimhan和Ramanan [11J，[12 J]）。(2)R.哈茨霍恩解决了以下问题：设E是p3上秩为2的半稳定向量丛。如果m是整数，使得X（E（m）>>且m> 0，则HO（p 3，E（m）>> 0。(3)S.兰顿利用这种运算证明了半稳定层模空间的适当性的”赋值准则”（[5 J]）。(4)正如我在上面提到的，这对于构造许多向量丛是有用的（Maruyama [6J和本文的§ 3）。(5)利用这一点，我们可以确定p2上的半稳定向量丛E的跳跃线曲线上一点的重数，其中r（E）= 2，cl（E）（见本文§ 4）。在这个时候，简单地说明一下初等变换似乎是有用的。在§ 1中，我将试图说明初等变换是什么。§ 2致力于说明
Introduction. About ten years ago I raised the following problem:" Construct many vector bundles on a higher dimensional projective variety". While considering this question, I found an operation on algebraic vector bundles, an elementary transformation. By using this operation, many vector bundles were constructed on every projective nonsingular variety. On the other hand, it appears recently that the transformation is a powerful tool in various directions of the theory of algebraic vector bundles. Let me mention some examples:(1) This played a key role in the deformation theory and the desingularization of the moduli spaces of vector bundles on curves (Narasimhan and Ramanan [llJ,[12J).(2) This has been used in essential way by R. Hartshorne to solve the following problem: Let E be a semi-stable vector bundle of rank 2 on p3. If m lS an integer such that x (E (m»> and m 0, then HO (p3, E (m» 0.(3) S. Langton exploited this operation to prove" valuative criterion" of the properness of moduli spaces of semi-stable sheaves ([5J).(4) As I mentioned in the above, this is useful to construct many vector bundles (Maruyama [6J and § 3 of this article).(5) By using this we can determine the multiplicity at a point of the curve of jumping lines of a semi-stable vector bundle E on p2 with r (E)= 2 and cl (E)(see § 4 of this article). It seems useful at this moment to give an expository account of elementary transformations. In § l I will try to show what the elementary transformation is.§ 2 is devoted to show several properties of