Let C be an algebraic curve of genus g ≥ 2. 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 a. We study the geometry of the moduli space of coherent systems for different values of a when k ≤ n and theudvariation of the moduli spaces when we vary a. As a consequence, for sufficiently large , we compute the Picard groups and the first and second homotopy groupsudof the moduli spaces of coherent systems in almost all cases, describe the moduli space for the case k = n − 1 explicitly, and give the Poincare polynomials for theudcase k = n − 2. In an appendix, we describe the geometry of the “flips” which takeudplace at critical values of a in the simplest case, and include a proof of the existenceudof universal families of coherent systems when GCD(n, d, k)= 1.
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机译：令C为g≥2的代数曲线。C上的相干系统由一对（E，V）组成，其中E是C在n级和d阶的C上的代数向量束，V是维数k的子空间相干系统的稳定性取决于参数a。我们研究了当k≤n时，对于a的不同值，相干系统的模空间的几何形状；当a改变时，模空间的变化。结果，对于足够大的情况，我们计算几乎所有情况下相干系统模空间的Picard群以及第一同构群和第二同构群ud，并明确描述k = n − 1情况下的模空间，并给出 udcase k = n − 2的Poincare多项式。在附录中，我们描述了“翻转”的几何形状，该翻转在最简单的情况下以a的临界值代替，并包括存在证明的 udof通用族当GCD（n，d，k）= 1。
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