Our purpose in this paper is to construct new examples of twisted Brill–Noether loci on curves of genus [Formula: see text]. Many of these examples have negative expected dimension. We deduce also the existence of a new region in the Brill–Noether map, whose points support non-empty standard Brill–Noether loci.

Let C be a smooth irreducible complex projective curve of genus $$g \ge 2$$ and M the moduli space of stable vector bundles on C of rank n and degree d with $$\gcd (n,d)=1$$ . A generalised Picard sheaf is the direct image on M of the tensor product of a universal bundle on $$M\times C$$ by the pullback of a vector bundle $$E_0$$ on C. In this paper, we investigate the stability of generalised Picard sheaves and, in the case where these are locally free, their deformations. When $$g\ge 3$$ , $$n\ge 2$$ (with some additional restrictions for $$g=3,4$$ ) and the rank and degree of $$E_0$$ are coprime, this leads to the construction of a fine moduli space for deformations of Picard bundles.

This article presents a list of open questions on higher rank Brill-Noether theory and coherent systems. Background material and appropriate references are included.

It is well known that there are no stable bundles of rank greater than 1 on the projective line. In this paper, our main purpose is to study the existence problem for stable coherent systems on the projective line when the number of sections is larger than the rank. We include a review of known results, mostly for a small number of sections.

Let $V$ be a vector bundle over a smooth curve $C$. In this paper, we study twisted Brill--Noether loci parametrising stable bundles $E$ of rank $n$ and degree $e$ with the property that $h^0 (C, V \otimes E) \ge k$. We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the Brill-Noether loci are nonempty, and in many cases have a component which is generically smooth and of the expected dimension. Along the way, we prove the irreducibility of certain components of both twisted and "nontwisted" Brill--Noether loci. We describe the tangent cones to the twisted Brill-Noether loci. We end with an example of a general bundle over a general curve having positive-dimensional twisted Brill--Noether loci with negative expected dimension.

Let $X$ be a smooth complex projective curve, and let $E$ be a vector bundle on $X$ which is not semistable. For a suitably chosen integer $r$, let $\text{Gr}(E)$ be the Grassmann bundle over $X$ that parametrizes the quotients of the fibers of $E$ of dimension $r$. Assuming some numerical conditions on the Harder-Narasimhan filtration of $E$, we study Seshadri constants of ample line bundles on $\text{Gr}(E)$. In many cases, we give the precise value of Seshadri constant. Our results generalize various known results for ${\rm rank}(E)=2$.

Let [Formula: see text] be a general generated coherent system of type [Formula: see text] on a general non-singular irreducible complex projective curve. A conjecture of D. C. Butler relates the semistability of [Formula: see text] to the semistability of the kernel of the evaluation map [Formula: see text]. The aim of this paper is to obtain results on the existence of generated coherent systems and use them to prove Butler’s Conjecture in some cases. The strongest results are obtained for type [Formula: see text], which is the first previously unknown case.

Higher rank Brill-Noether theory for genus 6 is especially interesting as, even in the general case, some unexpected phenomena arise which are absent in lower genus. Moreover, it is the first case for which there exist curves of Clifford dimension greater than 1 (smooth plane quintics). In all cases, we obtain new upper bounds for non-emptiness of Brill-Noether loci and construct many examples which approach these upper bounds more closely than those that are well known. Some of our examples of non-empty Brill-Noether loci have negative Brill-Noether numbers.

In this paper, we construct some examples of rank-2 Brill–Noether loci with “unexpected” properties on general curves. The key examples are in genus 6, but we also have interesting examples in genus 5 and in higher genus. We relate some of our results to the recent proof of Mercat’s conjecture in rank 2 by Bakker and Farkas.

Let (E, V ) be a general generated coherent system of type (n, d, n+m) on a general non-singular irreducible complex projective curve. A conjecture of D. C. Butler relates the semistability of E to the semistability of the kernel of the evaluation map V ⊗ OX → E. The aim of this paper is to obtain results on the existence of generated coherent systems and use them to prove Butler’s Conjecture in some cases. The strongest results are obtained for type (2, d, 4), which is the first previously unknown case.

Higher rank Brill-Noether theory is completely known for curves of genus $\leq 3$. In this paper, we investigate the theory for curves of genus 4. Some of our results apply to curves of arbitrary genus.

Let [Formula: see text] be a smooth projective complex curve of genus [Formula: see text]. We investigate the Brill–Noether locus consisting of stable bundles of rank 2 and determinant [Formula: see text] of odd degree [Formula: see text] having at least [Formula: see text] independent sections. This locus possesses a virtual fundamental class. We show that in many cases this class is nonzero, which implies that the Brill–Noether locus is nonempty. For many values of [Formula: see text] and [Formula: see text] the result is best possible. We obtain more precise results for [Formula: see text]. Appendix A contains the proof of a combinatorial lemma which we need.

In this paper, we consider higher rank Brill–Noether theory for smooth curves of genus 5, obtaining new upper bounds for non-emptiness of Brill–Noether loci and many new examples.

Let C be a smooth irreducible projective curve of genus g and L a line bundle of degree d generated by a linear subspace V of H0(L) of dimension n + 1. We prove a conjecture of D. C. Butler on the semistability of the kernel of the evaluation map V ⊗ 𝒪C → L and obtain new results on the stability of this kernel. The natural context for this problem is the theory of coherent systems on curves and our techniques involve wall crossing formulae in this theory.

In this paper we study G-Higgs bundles over an elliptic curve when the structure group G is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable G-Higgs bundles of a given topological type over an elliptic curve and we give an explicit description of the associated moduli space as a finite quotient of a product of copies of the cotangent bundle of the elliptic curve. We construct a bijective morphism from this new moduli space to the usual moduli space of semistable G-Higgs bundles, proving that the former is the normalization of the latter. We also obtain an explicit description of the Hitchin fibration for our (new) moduli space of G-Higgs bundles and we study the generic and non-generic fibres.

Over the past 20 years, a great deal of work has been done on the moduli spaces of coherent systems on algebraic curves. Until recently, however, there has been very little work on the fixed determinant case, except for the special case of rank 2 and canonical determinant. This situation has changed due to two papers of Osserman, who has obtained lower bounds for the dimensions of the fixed determinant moduli spaces in some cases. Our object in this paper is to show that some of Osserman's bounds are sharp.