Geometry of vector bundle extensions and applications to a generalised theta divisor

Let E and F be vector bundles over a complex projective smooth curve X, and suppose that 0 -> E -> W -> F -> 0 is a nontrivial extension. Let G be a subbundle of F, and D an effective divisor on X. We give a criterion for the subsheaf G(-D) \subset F to lift to W, in terms of the geometry of a scroll in the extension space \PP H^1 (X, Hom(F, E)). We use this criterion to describe the tangent cone to the generalised theta divisor on the moduli space of semistable bundles of rank r and slope g-1 over X, at a stable point. This gives a generalisation of a case of the Riemann-Kempf singularity theorem for line bundles over X. In the same vein, we generalise the geometric Riemann-Roch theorem to vector bundles of slope g-1 and arbitrary rank.Comment: Main theorem slightly weakened; statement and proof significantly more compac

Brill--Noether loci on moduli spaces of symplectic bundles over curves

The symplectic Brill--Noether locus ${\mathcal S}_{2n, K}^k$ associated to a curve $C$ parametrises stable rank $2n$ bundles over $C$ with at least $k$ sections and which carry a nondegenerate skewsymmetric bilinear form with values in the canonical bundle. This is a symmetric determinantal variety whose tangent spaces are defined by a symmetrised Petri map. We obtain upper bounds on the dimensions of various components of ${\mathcal S}_{2n, K}^k$. We show the nonemptiness of several ${\mathcal S}_{2n, K}^k$, and in most of these cases also the existence of a component which is generically smooth and of the expected dimension. As an application, for certain values of $n$ and $k$ we exhibit components of excess dimension of the standard Brill--Noether locus $B^k_{2n, 2n(g-1)}$ over any curve of genus $g \ge 122$. We obtain similar results for moduli spaces of coherent systems.Comment: Several references and acknowledgement added. 30 p

Moduli of symplectic bundles over curves

Let Х be a complex projective smooth irreducible curve of genus g. We begin by giving background material on symplectic vector bundles and principal bundles over X and introduce the moduli spaces we will be studying, In Chapter 2 we describe the stable singular locus and semistable boundary of the moduli space Mx(Sp2 C) of semistable principal Sp2 C-bundles over X. In Chapter 3 we give results on symplectic extensions and Lagrangian subbundles. In Chapter 4, we assemble some results on vector bundles of rank 2 and degree 1 over a curve of genus 2, which are needed in what follows. Chapter 5 describes a generically finite cover of Aix(Sp2C) for a curve of genus 2. In the last chapter, we give some results on theta-divisors of rank 4 symplectic vector bundles over curves: we prove that the general such bundle over a curve of genus 2 possesses a theta-divisor, and characterise those stable bundles with singular theta-divisors. Many results on symplectic bundles admit analogues in the orthogonal case, which we have outlined where possible