The aim of this paper is to generalize the [Formula: see text]-Segre invariant for vector bundles to coherent systems. Let [Formula: see text] be a non-singular irreducible complex projective curve of genus [Formula: see text] and [Formula: see text] be the moduli space of [Formula: see text]-stable coherent systems of type [Formula: see text] on [Formula: see text]. For any pair of integers [Formula: see text] with [Formula: see text], [Formula: see text] we define the [Formula: see text]-Segre invariant, and prove that it defines a lower semicontinuous function on the families of coherent systems. Thus, the [Formula: see text]-Segre invariant induces a stratification of the moduli space [Formula: see text] into locally closed subvarieties [Formula: see text] according to the value [Formula: see text] of the function. We determine an above bound for the [Formula: see text]-Segre invariant and compute a bound for the dimension of the different strata [Formula: see text]. Moreover, we give some conditions under which the different strata are nonempty. To prove the above results, we introduce the notion of coherent systems of subtype [Formula: see text].

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.

The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a nonsingular irreducible complex surface, and let E be a vector bundle of rank n on X. We use the m-elementary transformation of E at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;\,c_1,c_2)$ to $Hilb_{\, \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$ .

We extend the concept of the Segre invariant to vector bundles on a surface X. For X = ℙ2 we determine what numbers can appear as the Segre invariant of a rank 2 vector bundle with given Chern classes. The irreducibility of strata with fixed Segre invariant is proved and their dimensions are computed. Finally, we present applications to the Brill–Noether Theory for rank 2 vector bundles on ℙ2.