Collaborative Research: Vector Bundles on Projective Spaces

研究者和他的同事研究了射影空间上的小秩向量束。在有限特性和零特性下都得到了新的简化结构。在二阶束的情况下，这些构造在射影四空间上仅对正特征有效。研究者和他的同事们致力于将这些结构扩展到特征零的问题。这与这些束的变形理论有关。他们研究了这些束是否可以从正特征变形到零特征。这种束的变形理论也适用于关于线束的退化和和Hilbert格式的奇异分量的存在的问题。研究者和他的同事给出了被称为向量束的对象的明确构造。矢量束是编码大量几何图形（如曲线和面）的代数信息的设备。有了矢量束的明确知识，研究者和他的同事们就可以构建具有所需属性的几何图形。这项工作是用矩阵来完成的，它很容易在计算机代数系统中实现。大部分工作都是在有限的领域内完成的，这使得使用计算机可以给出精确的答案。在现实世界中涉及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。该项目研究存在于现实世界（特征为零的域上）的几何对象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.