. 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.

中文翻译：

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关于表面上的两个稳定束的注释

. 让π：X→C是在曲线上具有完整纤维的纤维化C并考虑极化H在表面上X. 让乙是一个秩为 2 的稳定向量丛C. 我们证明了回调π*(乙) 是一个H-稳定的捆绑X. 这个结果使我们能够关联稳定束的相应模空间 $${{\mathcal M}_C}(2,d)$$ 和 $${{\mathcal M}_{X,H}}(2,df,0)$$ 通过单射态射。我们研究了 Brill-Noether 基因座水平的诱导态射，以构建纤维表面上的 Brill-Noether 基因座的例子。关于 Brill-Noether 轨迹空性的结果是 Clifford 定理对曲面上的二阶丛的推广的结果。