Collaborative Research: Vector Bundles on Projective Spaces

合作研究：射影空间上的向量丛

• 批准号: 0070438
• 负责人: Christopher Peterson
• 金额: $ 5.47万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070438&HistoricalAwards=false
• 关键词: Collaborative; Research; Vector; Bundles; Projective

abstractThe investigator and his colleagues study small rank vector bundles on projective spaces. New and simplified constructions are obtained both in finite and in zero characteristics. In the case of rank two bundles, these constructions on projective four space are valid only in positive characteristics. The investigator and his colleagues work on the question of extending these constructions to characteristic zero. This is related to the deformation theory of these bundles. They investigate whether these bundles can be deformed from positive to zero characteristic. The deformation theory of such bundles also has applications to questions regarding degenerating sums of line bundles and the existence of exotic components of the Hilbert scheme.The investigator and his colleagues give explicit constructions of objects called vector bundles. Vector bundles are devices which encode algebraic information about huge numbers of geometric figures like curves and surfaces. With their explicit knowledge of vector bundles, the investigator and his colleagues can then construct geometric figures with desired properties. The work is done using matrices which are easily implemented on computer algebra systems. Much of the work is done over finite fields which allows the use of computers to give exact answers. Problems involving geometric modeling in the real world require approximating the answers, one method being the use of such finite fields. The project studies the interplay between geometric objects existing in the real world (over fields of characteristic zero) and their approximations over finite fields.

研究者和他的同事研究了射影空间上的小秩向量丛。 新的和简化的结构得到有限和零特征。 在秩二丛的情况下，这些在射影四空间上的构造仅在正特征上有效。 研究者和他的同事致力于将这些结构扩展到特征零的问题。 这与这些丛的变形理论有关。 他们研究这些束是否可以从正特性变形为零特性。 这种丛的变形理论也有应用的问题，关于退化和线丛和存在的奇异组件的希尔伯特scheme.The调查员和他的同事给出明确的建设对象称为向量丛。 向量束是一种编码大量几何图形（如曲线和曲面）的代数信息的装置。 有了向量丛的明确知识，研究者和他的同事就可以构造具有所需性质的几何图形。 这项工作是使用矩阵很容易实现的计算机代数系统。 大部分工作是在有限域上完成的，这使得计算机能够给出精确的答案。 涉及真实的世界中的几何建模的问题需要近似的答案，一种方法是使用这样的有限域。 该项目研究存在于真实的世界（特征为零的领域）的几何对象之间的相互作用及其在有限领域的近似。