A note on rank two stable bundles over surfaces

TL;DR

This paper demonstrates that stable rank two bundles on a curve remain stable when pulled back to a fibered surface, establishing links between their moduli spaces and analyzing Brill-Noether loci through a generalized Clifford's theorem.

Contribution

It proves the stability of pulled-back bundles on fibered surfaces and relates their moduli spaces, also exploring Brill-Noether loci and a generalized Clifford's theorem for rank two bundles.

Findings

Pullback of stable bundles remains stable on fibered surfaces.

Injective morphism between moduli spaces of bundles on curves and surfaces.

Criteria for emptiness of Brill-Noether loci based on generalized Clifford's theorem.

Abstract

Let $\pi: X \longrightarrow C$ 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 $\pi^*E$ 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.