George Hitching (Hannover)
Title: Rank 4 symplectic vector bundles on a curve of genus 2 without theta
divisors
Abstract: Let X be a complex projective smooth curve of genus g > 1 and W a
vector bundle of trivial determinant over X. We consider the set S(W) of line
bundles L of degree g-1 over X such that (L \otimes W) has nonzero sections.
For general W this is the support of a divisor on the (g-1)st Jacobian of X,
called the theta divisor of W. However, it may happen that S(W) is the whole
Jacobian; such bundles have been studied by Raynaud, Popa and others.
Now a vector bundle W is called symplectic if there is an antisymmetric
isomorphism between W and its dual. When X has genus 2, we construct a rank 4
symplectic bundle without a theta divisor, and conjecture that there are only
finitely many examples in this case.
This work was begun in the speaker's recent Ph. D. thesis with Christian
Pauly.