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 subset of F be a subbundle and D an effective divisor on X. We give a criterion for the subsheaf G(-D) subset of F to lift to W, in terms of the geometry of a scroll in the extension space PH1 (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.