Collaborative Research: Vector Bundles on Projective Spaces

研究人员和他的同事研究了射影空间上的小秩向量丛。得到了有限特征和零特征的新结构和简化结构。在秩二丛的情况下，投射四空间上的这些构造只在正特征中有效。这位研究人员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。他们调查这些丛是否可以从正特征变形到零特征。这类丛的形变理论也可以应用到线丛的退化和以及Hilbert模式的奇异分量的存在性等问题上。矢量束是对大量几何图形(如曲线和曲面)的代数信息进行编码的装置。有了向量丛的明确知识，研究人员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完成的，这些矩阵很容易在计算机代数系统上实现。许多工作都是在有限域上完成的，这使得计算机可以给出准确的答案。现实世界中涉及几何建模的问题需要近似解答，其中一种方法是使用这样的有限域。该项目研究现实世界中存在的几何对象(在特征为零的域上)与它们在有限域上的近似之间的相互作用。

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.