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。对于无法用其他方法得到的个别数学对象，通常可以通过将它们视为相似对象族的成员，即模空间中的点来证明。重要的例子给出了模空间的曲线，这使深入研究光滑曲线和退化，是一个原型的研究人员研究模空间的高维品种。此外，由于曲线出现在如此多的背景下，曲线的模空间是一个共同的交汇点，以基本的方式与数学和数学物理的许多不同领域相联系。这是一个研究使用顶点算子代数表示定义的尖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，也被称为共形块的向量丛，它在理解曲线的模空间的双有理几何中发挥了重要作用，特别是在亏格为零的曲线的情况下，它们被称为全局生成。具体地说，本研究将扩充定义在曲线模空间上的具有良好组合性质的向量丛类，用向量丛的形式找到模空间上无基点类的几何解释，这个奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准。

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.