Geometry of Vector Bundle Extensions and Applications to a Generalised Theta Divisor - Hitching, George H. (MATHEMATICA SCANDINAVICA, 2013)
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 F be a subbundle and D an effective divisor on X. We give a criterion for the subsheaf G(-D) F to lift to W, in terms of the geometry of a scroll in the extension space P 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.