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 $G \subseteq F$ be a subbundle 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 ${P} H^{1}(X, Hom(F, E))$. 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.