Conformal blocks and positive cycles on the moduli space of curves

曲线模空间上的共形块和正循环

• 批准号: 1201268
• 负责人: Angela Gibney
• 金额: $ 11.06万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201268&HistoricalAwards=false
• 关键词: Conformal; blocks; positive; cycles; moduli

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.

亏格g的稳定n点曲线的模空间具有有趣的正性性质和与数学和数学物理的许多不同领域有关的向量丛。在一点上，丛自然地被描述为共形块，即作为有理共形场理论中的基本对象而出现的矢量空间。这个提案中的项目集中在三个一般主题上，旨在加深我们对保形块丛的向量丛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陈氏班级的所谓“关键级别捆绑”受制于“奇怪的身份”，由级别和级别的互换角色赋予。第二个项目探索共形块因子的性质，包括它们对应的态射、有限生成问题和应用。第三个方案是关于簇上正循环的锥体，通过共形块来描述稳定的n点有理曲线的模空间，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显著的地位。作为模空间，它们深入到光滑曲线及其退化的研究中，作为特殊的变种，它们在发展一般理论方面发挥了重要作用。最近的工作表明，空间的许多组合方面是由共形块向量丛所体现的基本几何结构的反映。该方案的目的既是利用这些向量丛的特征来发现模空间的性质，也是利用模空间的体系结构来揭示向量丛之间的潜在关系。