  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.
