Let M (n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M (n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms M or P (G, M (n, ξ)) and the moduli space of stable bundles over X × G, where G is the Grassmannian G(n − r, C n ). Moreover, we give sufficient conditions for M or 2ns (P 1 , M (n, ξ)) to be non-empty, when s ≥ 1.

Let [Formula: see text] be a general generated coherent system of type [Formula: see text] on a general non-singular irreducible complex projective curve. A conjecture of D. C. Butler relates the semistability of [Formula: see text] to the semistability of the kernel of the evaluation map [Formula: see text]. The aim of this paper is to obtain results on the existence of generated coherent systems and use them to prove Butler’s Conjecture in some cases. The strongest results are obtained for type [Formula: see text], which is the first previously unknown case.

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We review the notion of Gieseker stability for torsion-free Higgs sheaves. This notion is a natural generalization of the classical notion of Gieseker stability for torsion-free coherent sheaves. In this article we prove some basic properties that are similar to the classical ones for torsion-free coherent sheaves over projective algebraic manifolds. In particular, we show that Gieseker stability for torsion-free Higgs sheaves can be defined using only Higgs subsheaves with torsion-free quotients; we also prove that a direct sum of two Higgs sheaves is Gieseker semistable if and only if the Higgs sheaves are both Gieseker semistable with equal normalized Hilbert polynomial; then we prove that a classical property of morphisms between Gieseker semistable sheaves also holds in the Higgs case; as a consequence of this and because of an existing relation between Mumford-Takemoto stability and Gieseker stability for Higgs sheaves, we obtain certain properties concerning the existence of Hermitian-Yang-Mills metrics, simplesness and extensions. Finally, we make some comments about Jordan-Hölder and Harder-Narasimhan filtrations for Higgs sheaves.