Ropes supported on a line embedded in a Grassmannian variety

Let L be a line contained in a Grassmannian variety G. A d-rope C ⊂ G sup- ported on L is a locally Cohen-Macaulay curve of degree d with Cred = L and (IL,G)2 ⊂ IC,G. We characterize the d-ropes C supported on L and embedded in G. In some cases we describe also the vector bundles on such a rope C. Finally, we describe the parameter spaces for ropes embedded in G.