Generalized Verlinde Bundles and Moduli Spaces of Curves

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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.