Subvarieties of -divisors in the Jacobian

We explore some of the interplay between Brill-Noether subvarieties of the moduli space of rank 2 bundles with canonical determinant on a smooth projective curve and -divisors, via the inclusion of the moduli space into , singular along the Kummer variety. In particular we show that the moduli space contains all the trisecants of the Kummer and deduce that there are quadrisecant lines only if the curve is hyperelliptic; we show that for generic curves of genus , though no higher, bundles with sections are cut out by ; and that for genus 4 this locus is precisely the Donagi-Izadi nodal cubic threefold associated to the curve.

     

Spin Verlinde Spaces and Prym Theta Functions

It is shown that the theta functions of level n on the principallypolarised Prym varieties of an algebraic curve are dual to thesections of the orthogonal theta line bundle on the moduli spaceof Spin(n)-bundles over the curve. As a by-product of our computations,we also note that when n is odd, the Pfaffian line bundle onmoduli space has a basis of sections labelled by the even thetacharacteristics of the curve. 1991 Mathematics Subject Classification:14D20, 14H42, 14H60, 14K25.     

Reciprocity Laws in the Verlinde Formulae for the Classical Groups

The Verlinde formula is computed for each of the simply-connected classical Lie groups, and it is shown that the resulting formula obeys certain reciprocity laws with respect to the exchange of the rank and the level. Some corresponding dualities between spaces of sections of theta line bundles over moduli spaces of -bundles on curves are conjectured but not proved.