Moduli of pairs and generalized theta divisors

Moduli of Brill-Noether Pairs on Algebraic Curves

“…For E T on T × X and Λ T on T , we shall exploit the natural isomorphism Hom(π * Λ T , E T ) ∼ = Hom(Λ T , π * E T ) to regard a family of Brill-Noether pairs as specified by a map j T : Λ T → π * E T such that, for each t ∈ T , E t is torsion-free and the induced map Λ t → H 0 (E t ) is injective. This last condition implies that j T is a sheaf injection, and we thus recover Definition 1.11 of [13]. Now the family on R ss comes from (i) the universal quotient sheaf E Q on Q × X, for which π * E Q can be identified with a subsheaf of V ⊗O Q over the good points corresponding to torsion-free sheaves, and (ii) the universal subsheaf L G ⊂ V ⊗O G on G. More accurately, let E R ss be the pull-back of E Q to R ss × X and let Λ R ss be the pull-back of L G to R ss .…”

“…Such pairs are the common generalisation of the classical notion of a linear system, for which E is (the sheaf of sections of) a line bundle, and of Bradlow pairs [3], for which one may take Λ to be 1-dimensional -although strictly speaking Λ should be replaced by a single non-zero section φ. These more general pairs have also been studied by Bertram [2], Raghavendra and Vishwanath [13] (under the name of "pairs") and by Le Potier [8] (under the name of "coherent systems"), who works over varieties (or schemes) of arbitrary dimension. In this paper, we shall use the term "Brill-Noether pair", to emphasise the role that we hope they will play in higher rank Brill-Noether theory.…”

“…In higher rank, one must introduce a notion of semistability, which depends on a single parameter α (see Definition 2.3.2). The same definition is used in [13], while in [3] an equivalent definition is used involving τ = d+αl r . The result we shall prove is the following.…”

Singularities of Brill‐Noether loci for vector bundles on a curve

“…In this section we review certain facts about coherent systems which we will need in what follows. We refer the reader to Bradlow et al 3, King‐Newstead 18, He 14, Le Potier 24 and Raghavendra‐Vishwanath 29 for more details. Roughly, a coherent system of type ( k , d , r ) is a pair ( E , V ) with E a vector bundle of rank r and degree d and V ∈ G ( k , H 0 ( E )).…”

Fourier-Mukai transforms for coherent systems on elliptic curves

“…The full subcategory S α,μ (X) of C(X) consisting of α-semistable coherent systems with fixed α-slope μ is a Noetherian and Artinian abelian category with simple objects that are precisely the α-stable coherent systems (see [15,23]).…”