Uniformization of the Moduli Space of Pairs Consisting of a Curve and a Vector Bundle

TL;DR

This paper investigates the uniformization of the moduli space of algebraic curves with vector bundles by analyzing a related space of tuples and establishing a transitive group action and Lie algebra identities.

Contribution

It introduces a new framework for understanding the uniformization of moduli spaces of curves and vector bundles via group actions and Lie algebra identities.

Findings

Identified a group acting transitively on the moduli space of tuples.

Proved an identity between central extensions of Lie algebras.

Provided a geometric explanation for the Lie algebra identity.

Abstract

This paper is devoted to the study of the uniformization of the moduli space of pairs (X, E) consisting of an algebraic curve and a vector bundle on it. For this goal, we study the moduli space of 5-tuples (X, x, z, E, \phi), consisting of a genus g curve, a point on it, a local coordinate, a rank n degree d vector bundle and a formal trivialization of the bundle at the point. A group acting on it is found and it is shown that it acts (infinitesimally) transitively on this moduli space and an identity between central extensions of its Lie algebra is proved. Furthermore, a geometric explanation for that identity is offered.