Some moduli spaces of $\alpha$-stable coherent systems on algebraic surfaces

Let $X$ be a smooth, irreducible, projective algebraic surface, and let $\alpha \in \mathbb{Q}[m{]}_{>0}$ be a polynomial. In this paper, we determine topological and geometric properties of the moduli space of $\alpha$-stable coherent systems of type $(n;c_1,c_2,k)$ with $k<n$ on $X$, for sufficiently large values of $\alpha$. We prove that, for $\alpha$ sufficiently large, the moduli space admits a description as a Grassmann bundle over a moduli space of $H$-stable torsion free sheaves. As a consequence, we obtain results on irreducibility, dimension. Our approach relies on establishing a correspondence between $\alpha$-stable coherent systems and extensions of $H$-stable torsion free sheaves by trivial bundles.