Generalized Verlinde Bundles and Moduli Spaces of Curves

模空间揭示了特定类型的对象在族中的行为方式。关于单个数学对象的结果，如果用其他方法无法达到，通常可以通过将它们视为一族相似对象的成员来证明，即，作为模空间中的点。给出了曲线的模空间的重要例子，为光滑曲线及其退化的研究提供了新的思路，也是研究高维变元的模空间的原型。此外，由于曲线在如此多的背景下出现，曲线的模空间是一个公共的交汇点，它以一种基本的方式与许多不同的数学和数学物理领域联系在一起。这是一个研究使用顶点算子代数的表示定义的点曲线的模空间上的层的项目。这些层提供了Verlinde丛的自然推广，也称为共形块的向量丛，它们在理解曲线的模空间的双曲线几何方面发挥了重要作用，特别是在亏格为零的曲线的情况下，其中已知它们是全局生成的。具体地说，这项研究将扩大定义在曲线的模空间上的具有良好组合性质的向量丛的类别，找到这些模空间上基点自由类的几何解释，并使用所得到的信息来刻画这些模空间的态射。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Moduli spaces reveal how objects of a particular type behave in families. Results about individual mathematical objects that are unreachable by other means can often be proved by considering them as members of a family of similar objects, that is, as points in a moduli space. Important examples are given by moduli spaces of curves, which give insight into the study of smooth curves and their degenerations, and are a prototype for researchers studying moduli spaces of higher dimensional varieties. Moreover, as curves arise in so many contexts, moduli spaces of curves are a common meeting ground, connected in a fundamental way with many disparate fields of mathematics and mathematical physics.This is a project to study sheaves on moduli spaces of pointed curves defined using representations of vertex operator algebras. These sheaves provide a natural generalization of Verlinde bundles, also known as vector bundles of conformal blocks, which have played an important role in understanding the birational geometry of moduli spaces of curves, particularly in the case of curves of genus zero, where they are known to be globally generated. Specifically, this research will enlarge the class of vector bundles with good combinatorial properties defined on moduli spaces of curves, find geometric interpretations for basepoint free classes on these moduli spaces in terms of vector bundles, and use the resulting information to characterize morphisms to these moduli spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.