The Equivalence of Hilbert and Mumford Stability for Vector Bundles

Abstract

In this paper, we prove that the notions of Hilbert stability and Mumford stability agree for vector bundles of arbitrary rank over smooth curves. The notion of Hilbert stability was introduced by Gieseker and Morrison in 1984, and they showed that for smooth curves and vector bundles of rank two it agrees with Mumford stability. A different proof for the rank two case was given by M. Teixidor i Bigas. Our proof uses a new approach and avoids complicated computations. Our results might serve as a first step in the construction of the Hilbert stable compactification of the universal moduli space of stable vector bundles over the moduli space of smooth curves as suggested by Teixidor.