Geometry of vector bundle extensions and applications to the generalised theta divisor

Details Geometry of vector bundle extensions and applications to the generalised theta divisor Geometry of vector bundle extensions and applications to the generalised theta divisor

2006-10-31

arxiv.org/abs/math/0610970v3

Mathematics

Algebraic Geometry

32 pages; some discussion added, mistake in Theorem 4 corrected

Scientific paper

Let E and F be vector bundles over a smooth curve X, and suppose that 0 -> E -> W -> F -> 0 is a nontrivial extension. Suppose \gamma : G -> F is a vector bundle map which is generically injective. We give a criterion for \gamma to factorise via W in terms of the geometry of a projective bundle in the extension space \PP H^1 (X, Hom(F,E)). We use this 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 give a generalisation of the geometric Riemann-Roch theorem to vector bundles of arbitrary rank over X.

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