Brill-Noether loci and generated torsionfree sheaves over nodal and cuspidal curves

Higher rank Brill—Noether theory and coherent systems open questions

“…Most work has concentrated on the case of a curve with nodal (or possibly cuspidal) singularities. Brill-Noether theory on irreducible nodal and cuspidal curves has been studied, for example, in [9,15]. In particular, the results of [21] are generalised in [9], while kernel bundles are discussed in [15], including a proof of semistability (stability) in the case d ≥ 2ng (d > 2ng) (compare [32]); [15] also contains generalisations of the results of [76], [78].…”

“…Brill-Noether theory on irreducible nodal and cuspidal curves has been studied, for example, in [9,15]. In particular, the results of [21] are generalised in [9], while kernel bundles are discussed in [15], including a proof of semistability (stability) in the case d ≥ 2ng (d > 2ng) (compare [32]); [15] also contains generalisations of the results of [76], [78]. Coherent systems on integral curves are discussed in [6,7] and on irreducible nodal curves in [10], where results from [16] and [17] are generalised, including a description of G L (α; n, d, k) when k ≤ n. The case of a nodal curve of (arithmetic) genus 1 is discussed in [11].…”

Higher rank Brill-Noether theory and coherent systems: Open questions

“…Most work has concentrated on the case of a curve with nodal (or possibly cuspidal) singularities. Brill-Noether theory on irreducible nodal and cuspidal curves has been studied, for example, in [Bh07,BhS13]. In particular, the results of [BGN97] are generalised in [Bh07], while kernel bundles are discussed in [BhS13], including a proof of semistability (stability) in the case d ≥ 2ng (d > 2ng) (compare [Bu94]).…”

“…Brill-Noether theory on irreducible nodal and cuspidal curves has been studied, for example, in [Bh07,BhS13]. In particular, the results of [BGN97] are generalised in [Bh07], while kernel bundles are discussed in [BhS13], including a proof of semistability (stability) in the case d ≥ 2ng (d > 2ng) (compare [Bu94]). Coherent systems on integral curves are discussed in [Ba06b,BaP07] and on irreducible nodal curves in [Bh09], where results from [BG02] and [BGMN03] are generalised, including a description of G L (α; n, d, k) when k ≤ n. The case of a nodal curve of (arithmetic) genus 1 is discussed in [Bh11].…”

Hyperkähler varieties as Brill-Noether loci on curves

“…see[BS13, Proposition 4.3] for more details. Note that the proof in[BS13] is valid for any family of simple sheaves on a variety.The moduli space M X (v) is a (non-empty) smooth quasi-projective variety, see for instance [Huy16, Chapter 10, Corollary 2.1 & Theorem 2.7]. Hence to prove Theorem 1.1, we only need to show the derivative of the restriction map dΨ…”