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 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.
Cite
@article{arxiv.math/0610970,
title = {Geometry of vector bundle extensions and applications to a generalised theta divisor},
author = {George H. Hitching},
journal= {arXiv preprint arXiv:math/0610970},
year = {2013}
}
Comments
Main theorem slightly weakened; statement and proof significantly more compact