2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/100831Let $C$ 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 $α$. We study the geometry of the moduli space of coherent systems for different values of $α$ when $k\leq n$ and the variation of the moduli spaces when we vary $α$. As a consequence, for sufficiently large $α$, 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 Poincaré polynomials for the case $k=n-2$.38 pages; v3. Appendix and new references added; v4. minor corrections, two added references; v5. final version, one typo corrected and one reference deletedAlgebraic GeometryDifferential Geometry14D20; 14H51; 14H60text