Let C be a smooth irreducible complex projective curve of genus g and let B k (2, K C ) be the Brill-Noether locus parametrizing classes of (semi)stable vector bundles E of rank two with canonical determinant over C with h 0 (C, E) ≥ k. We show that B 4 (2, K C ) has an irreducible component B of dimension 3g −13 on a general curve C of genus g ≥ 8. Moreover, we show that for the general element [E] of B, E fits into an exact sequence 0 → O C (D) → E → K C (−D) → 0 with D a general effective divisor of degree three, and the corresponding coboundary map ∂ : H 0 (C, K C (−D)) → H 1 (C, O C (D)) has cokernel of dimension three.

Abstract. Let π : X → C be a fibration with integral 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 π*(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.

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