Conformal blocks and positive cycles on the moduli space of curves

g属的稳定n点曲线的模空间被赋予了具有有趣的正性质的向量束，并且与数学和数学物理的许多不同领域有联系。在一个点上，束自然地被描述为共形块，向量空间在有理共形场论中作为基本对象出现。本提案中的项目集中在三个一般主题上，旨在发展我们对共形块束的矢量束及其与模空间上的循环的相互作用的理解。第一个项目旨在展示所谓的“临界水平束”的Chern类服从于“奇怪的身份”，通过交换水平和等级的角色。第二个项目探讨共形块除数的性质，包括它们对应的态射、有限生成问题和应用。第三个项目是关于变量上正循环的锥，用保形块的稳定n点有理曲线的模空间来说明。具有标记点的稳定曲线的模空间在代数几何宇宙中占有独特而突出的地位。作为模空间，它们为光滑曲线及其退化的研究提供了新的思路，作为特殊的变体，它们在发展一般理论方面发挥了重要作用。最近的研究表明，空间的许多组合方面是由共形块的矢量束体现的底层几何结构的反射。本提案中的项目旨在利用这些向量束的特征来发现模空间的本质，并利用模空间的体系结构来揭示向量束之间的潜在关系。

The moduli space of stable n-pointed curves of genus g, is endowed with vector bundles having interesting positivity properties and connections to many different areas of mathematics and mathematical physics. Over a point, the bundles are naturally described as conformal blocks, vector spaces that arise as basic objects in rational conformal field theory. The projects in this proposal are focused on three general themes, aimed at developing our understanding of vector bundles of conformal blocks bundles and their interplay with cycles on the moduli space. The first project aims to show the Chern classes of so-called "critical level bundles" are subject to ''strange identities'', given by interchanging roles of level and rank. The second project explores properties of conformal blocks divisors, including their corresponding morphisms, questions of finite generation, and applications. The third project is concerned with cones of positive cycles on varieties, illustrated for the moduli space of stable n-pointed rational curves via conformal blocks.Moduli spaces of stable curves with marked points occupy a unique and prominent position in the algebro-geometric universe. As moduli spaces, they give insight into the study of smooth curves and their degenerations, and as special varieties, they have played an important role in developing general theory. Recent work has revealed that many combinatorial aspects of the spaces are reflections of underlying geometric structures embodied by vector bundles of conformal blocks. The projects in this proposal aim both to use the features of these vector bundles to discover the nature of the moduli spaces, and also to use the architecture of the moduli spaces to reveal underlying relationships between the vector bundles.