Spectral Curves, Theta Divisors and Picard Bundles

Prym varieties and integrable systems

“…Thus the orbit X 0 coincides with the deformation space N (s) ⊗ f * F , which is covered by Jac(C 0 ). The last statement of the theorem follows from a result of [13]. This completes the proof.…”

Stability of the Picard Bundle

“…These Picard bundles have been studied for n = 1 in [4,5] and for general n in [6], where it was proved that, for d > 2ng, the Picard bundle is stable with respect to the generalised theta divisor on M n,d . The argument depends on pulling W back to an open subset of the Jacobian of a spectral curve Y over X, and then using the Abel-Jacobi map Y → Pic 0 (Y ).…”

Singularities of Brill‐Noether loci for vector bundles on a curve

“…Recall that given \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathscr{U}_r=\mathscr{U}(r,r(g-1))$\end{document}, the subvariety B 1 r , r ( g − 1) is a divisor which we will denote Θ r . Given a general \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$E\in \mathscr{U}_r$\end{document}, it is known that the map defined by ξ↦ξ⊗ E is an embedding (see 25). It is easy to check by degenerating E to a direct sum of r line bundles that if \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\Theta _r|_{\rm {Pic}^0(C)}=\Theta _E$\end{document} is a divisor, then Θ E ∈ | r Θ|, where Θ is a translate of the Riemann theta divisor on \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\rm {Pic}^0(C)$\end{document}.…”