A note on rank two stable bundles over surfaces

Abstract

Let $\pi :X⟶C$ be a fibration with reduced fibers over a curve $C$ and consider a polarization $H$ on the surface $X$. Let $E$ be a stable vector bundle of rank $2$ on $C$. We prove that the pullback ${\pi }^{\ast }E$ is a $H-$stable bundle over $X$. This result allows us to relate the corresponding moduli spaces of stable bundles ${\mathcal{M}}_C(2,d)$ and ${\mathcal{M}}_{X,H}(2,df,0)$ through an injective morphism. We study the induced morphism at the level of Brill-Noether loci to construct examples of Brill-Noether loci on fibered surfaces. Results concerning the emptiness of Brill-Noether loci follow as a consequence of a generalization of Clifford's Theorem for rank two bundles on surfaces.

arXiv comment15 pages

References in corpus (1)

• Stable Higgs Bundles on ruled surfaces