Classification of Noncommutative Projective Surfaces

本项目的目标是在非对易射影曲面的分类方面取得进展。 这是一个重要的问题，因为许多有趣的新现象出现在曲面上，而曲线上却没有;因此，更好地理解曲面将促进我们对非对易几何的理解。 首先，PI计划完成双有理交换曲面的分类--这是超越度为2的交换函数场的非交换曲面--根据naivenon交换爆破的过程。 其次，我们想研究包含点的交换曲线的非交换曲面-我们将从研究一类可能与M有关的3维椭圆正则代数的子代数开始。货车den Bergh关于非对易爆破的另一个概念。PI计划研究胖点模，关于它有许多基础问题。最后，人们应该研究朴素爆破到高维非交换簇的推广。非交换射影几何是一个相对较新的学科，它试图使用一些与交换代数几何相同的直觉和技术，但在非交换设置。 现代物理学告诉我们，那些可能彼此不对易的操作的重要性，这意味着你执行它们的顺序很重要。 例如，著名的海森堡测不准原理的一种解释是位置和动量不对易。 在非交换代数和物理学之间还有许多其他的联系。 这个项目希望能更好地抽象理解代数中的某些结构（低维非交换分次环），其中包括一些著名的例子，如Sklyanin代数，这与物理世界的研究有关。

The objective of this project is to make progress on parts of theclassification of noncommutative projective surfaces. This is animportant problem, because many interesting new phenomena occur forsurfaces that are absent for curves; thus understanding the surfacecase better will advance our understanding of noncommutativegeometry in general. First, the PI plans to complete theclassification of birationally commutative surfaces---which arenoncommutative surfaces with a commutative function field oftranscendence degree 2---in terms of the process of naivenoncommutative blowing up. Next, one wants to study noncommutativesurfaces containing a commutative curve of points---we will start byexamining a class of subalgebras of elliptic regular algebras ofdimension 3, which may be related to M. Van den Bergh's differentnotion of noncommutative blowing up. The PI plans to study fat pointmodules, about which there are many foundational questions.Finally, one should study the generalization of naive blowing up tohigher dimensional noncommutative varieties.Noncommutative projective geometry is a relatively new subject,which attempts to use some of the same intuition and techniquesunderlying commutative algebraic geometry, but in the noncommutativesetting. Modern physics tells us the importance of consideringoperations that may not commute with each other, which means itmatters which order you perform them. For example, oneinterpretation of the celebrated Heisenberg Uncertainty Principle isthat position and momentum do not commute. There are many otherconnections between noncommutative algebra and physics. Thisproject hopes to gain a better abstract understanding of certainkinds of structures in algebra (noncommutative graded rings of lowdimension) which include among them some famous examples, such asthe Sklyanin algebras, which are related to the study of thephysical world.