Some applications of the theory of positive vector bundles
Some applications of the theory of positive vector bundles
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正向量丛理论的一些应用
DOI：
10.1007/bfb0099356
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发表时间：
1984
期刊：
影响因子：
--
通讯作者：
R. Lazarsfeld
中科院分区：
文献类型：
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作者：
R. Lazarsfeld
介绍一个相当大的机构的工作已经发展在过去几年中松散的积极性在代数几何的概念为中心。一方面，许多结果已经出现了什么可能被称为几何的射影空间，主题是往往显着的特殊性质享有低余维subvar让。[27][28][29][2这些结果依赖于射影空间本身的正性，例如在贝尔蒂尼类型的各种定理中所表现的。在另一个方向上，正向量丛的一般理论最近得到了扩展，更有趣的是，它被应用于各种几何情形（参见：[42]，[12]，[13]，[31]，[7]）。将这两组结果连接起来，在一个类中，Mori对Frankel-Hartshorne的意义深远的证明（[35]，[6]）。我们在CIME会议上的演讲主要涉及射影空间的几何，特别是FL Zak关于线性正规性的工作（[43]，[32]）。此外，我们还讨论了Z. Ran [391]与哈茨霍恩关于完全相交的猜想[27]有关。大部分这类材料已调查其他地方（cf. [23]，[11]，[32]），我们不打算在这里重复已有的文献，本文将构成我们可能在Acirea会议上讲过的一门课程的笔记，重点是正向量丛及其应用。我们从（§ 1）开始，对一般理论作一个基本的概述，强调高阶的线丛和向量丛的情况之间的相似性和不同性。其余部分致力于博览会的几个以前未发表的证据和结果。在§ 2中，我们给出了一个定理的简单拓扑证明，该定理保证在适当的正性和维数假设下，向量丛映射必须降秩。然后，我们描绘如何应用这一点，沿着[12]的路线，给出一个快速证明（轻微推广）最近的定理Ghione [16]关于存在的特殊因子相关联的向量丛代数曲线。在§ 3中，我们利用E）Goresky-MacPherson [17]的一个定理证明了某些正向量丛（Thm. 3.5）。我们
Introduction A considerable body of work has developed over the last few years loosely centered about the notion of positivity in algebraic geometry. On the one hand, numerous results have appeared on what might be called the geometry of projective space, the theme being the often remarkable special properties enjoyed by low codimensional subvar-Let. Les of, and mappings to, projective space (cr,[L},[27],[9],[ll],[151,[31],[43], f32]). These results depend on the positivity of projective Space itself, as manifested for example in various theorems of Bertini type. In another direction, the general theory of positive vector bundles has recently been extended and, more interestingly, applied in various geometric situations (cf.[42],[12],[13],[31],[7]). Bridging these two groups of results, in a class all by itself, one has Mori's far-reaching proof of the Frankel-Hartshorne([35],[6]). Our lectures at the CIME conference were largely concerned with the geometry of projective space, and especially with the work of FL Zak on linear normality ([43],[32]). In addition, we discussed a recent theorem of Z. Ran [391 related to Hartshorne's conjecture [27J on complete intersections. Most of this material has been surveyed eLsewhere (cf.[23],[11],[32]), and we do not propose to duplicate t he existing literature here.The present paper will rather constitute the notes to a course that we might have given at the Acirea1e conference, focusing on positive vector bundles and their applications. We start (§ 1) with an elementary overview of the general theory, emphasizing the similarities and differences between the cases of line bundles and vector bundles of higher rank. The remaining sections are devoted to expositions of several previously unpublished proofs and results. In § 2 we give a simple topological proof of a theorem guaranteeing that under suitable positivity and dimensional hypotheses a map of vector bundles must drop rank. We then sketch how this may be applied, along the lines of [12], to give a quick proof of (a slight generalization of) a recent theorem of Ghione [16] concerning the existence of special divisors associated to a vector bundle on an algebraic curve. In § 3 we usE) a theorem of Goresky-MacPherson [17J to prove a homotopy Lefschetz-type result for the zero-loci of sections of certain positive vector bundles (Thm. 3.5). We