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Vector bundles and theta functions on curves of genus 2 and 3
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 128, Number 3, June 2006
- pp. 607-618
- 10.1353/ajm.2006.0018
- Article
- Additional Information
Let SUC(r) be the moduli space of vector bundles of rank r and trivial determinant on a curve C. A general E in SUC(r) defines a divisor ΘE in the linear system |rΘ|, where Θ is the canonical theta divisor in Picg-1 (C). This defines a rational map Θ: SUC(r) → |rΘ|, which turns out to be the map associated to the determinant bundle on SUC(r) (the positive generator of Pic (SUC(r)). In genus 2 we prove that this map is generically finite and dominant. The same method, together with some classical work of Morin, shows that in rank 3 and genus 3 the theta map is a finite morphism - in other words, every vector bundle in SUC(3) admits a theta divisor.