Geometry of Vector Bundle Extensions and Applications to a Generalised Theta Divisor

Abstract

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 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.