On the geometry of moduli spaces of coherent systems on algebraic curves

Let $C$ be an algebraic curve of genus $g$. A coherent system on $C$ consists of a pair $(E,V)$, where $E$ is an algebraic vector bundle over $C$ of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of the space of sections of $E$. The stability of the coherent system depends on a parameter $\alpha$. We study the geometry of the moduli space of coherent systems for different values of $\alpha$ when $k\leq n$ and the variation of the moduli spaces when we vary $\alpha$. As a consequence, for sufficiently large $\alpha$, we compute the Picard groups and the first and second homotopy groups of the moduli spaces of coherent systems in almost all cases, describe the moduli space for the case $k=n-1$ explicitly, and give the Poincar\'e polynomials for the case $k=n-2$.