On $(k,l)$-stable vector bundles over algebraic curves

On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

“…The non-emptiness of A k,ℓ (n, ξ) is established in [16,Proposition 2.4]. The following proposition is a reformulation of [16,Proposition 2.4] and [20, Lemma 5.5] (see also [16]) in terms of r-elementary transformations.…”

“…In particular, if k and ℓ are non-negative integers, (k, ℓ)-stability implies stability. However, for negative values of k and ℓ, a (k, ℓ)-stable bundle does not need to be stable (see [16]). Denote by A k,ℓ (n, d) the set of (k, ℓ)-stable vector bundles of rank n and degree d over X and let A k,ℓ (n, ξ) := A k,ℓ (n, d) M(n, ξ).…”

“…In order to state our results we recall from [16] that using (k, ℓ)-stable bundles, one can generalise the idea of the good Hecke cycles to obtain the notion of Hecke Grassmannians (see §2) in M(n, ξ). It has recently been noted by O. Mata-Gutiérrez that through a very general point E ∈ M(n, ξ), there exists Hecke Grassmannians passing through E. The Hecke Grassmmannian is called an r-Hecke cycle, with 0 < r < n, if it defines a closed subscheme.…”