Subvarieties of Su C (2) and 2θ-divisors in the Jacobian

Let SUC(2, L) denote the projective moduli variety of semistable rank 2 vector bundles with determinant L ∈ Pic(C) on a smooth curve C of genus g > 2; and suppose that degL is even. It is well-known that, on the one hand, the singular locus of SUC(2, L) is isomorphic to the Kummer variety of the Jacobian; and on the other hand that when C is nonhyperelliptic SUC(2,O) has an injective morphism into the linear series |2Θ| on the Jacobian J C which restricts to the Kummer embedding a 7→ Θa + Θ−a on the singular locus. Dually SUC(2, K) injects into the linear series |L| on J C , where L = O(2Θκ) for any theta characteristic κ, and again this map restricts to the Kummer map J C → |2Θ| ∨ = |L| on the singular locus. This map to projective space (the two cases are of course isomorphic) comes from the complete series on the ample generator of the Picard group, and (at least for a generic curve) is an embedding of the moduli space. Moreover, its image contains much of the geometry studied in connection with the Schottky problem; notably the configuration of Prym-Kummer varieties. In this paper we explore a little of the interplay, via this embedding, between the geometry of vector bundles and the geometry of 2θ-divisors. On the vector bundle side we are principally concerned with the Brill-Noether loci W ⊂ SUC(2, K) defined by the condition h(E) > r on stable bundes E. These are analogous to the very classical varieties W r g−1 ⊂ J g−1 C . Unlike the line bundle theory, however, general results—connectedness, dimension, smoothness and so on—are not known for the varieties W (see [6]). On the 2θ side we shall consider the Fay trisecants of the 2θ-embedded Kummer variety, and the subseries PΓ00 ⊂ |L| consisting of divisors having multiplicity ≥ 4 at the origin. This subseries is known to be important in the study of principally polarised abelian varieties [10]: in the Jacobian of a