User aaron gerding - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:40:40Z http://mathoverflow.net/feeds/user/16298 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69754/pulling-back-a-line-bundle-on-the-jacobian-to-a-spin-bundle-on-the-curve Pulling back a line bundle on the Jacobian to a spin bundle on the curve aaron gerding 2011-07-07T22:35:56Z 2011-07-08T23:35:49Z <p>I'd like to have an expression for the (or some) line bundle on the Jacobian $J$ of a smooth complex projective curve $C$ with genus $g >1$ which pulls back to a chosen spin bundle (theta characteristic) $\kappa$ on $C$ via the Abel-Jacobi map $\alpha_c$ based at $c \in C$.</p> <p>I know (from looking at Birkenhake and Lange 11.3) that by Riemann's Theorem and the Seesaw principle </p> <p>$\alpha_c^*\mathcal{O}_J(\Theta _{\kappa}) = \kappa \otimes \mathcal{O}_C(c)$, </p> <p>where $\Theta _{\kappa}$ is the (symmetric) theta divisor that $\kappa$ determines by $\alpha _{\kappa}^* \Theta _{\kappa} = W _{g-1} \subset Pic^{g-1}$. So what I guess I'm looking for is a way to move the $c$ dependence over to the left side of this equation. For what it's worth, my hope is to be able to lift the vector bundle $\kappa \oplus \kappa^{-1}$ to $J$, which I'm assuming is just a matter of lifting $\kappa$.</p> <p>This doesn't seem difficult but I don't have a lot of experience with Abelian varieties and I'm not sure how to get started.</p> http://mathoverflow.net/questions/69754/pulling-back-a-line-bundle-on-the-jacobian-to-a-spin-bundle-on-the-curve Comment by aaron gerding aaron gerding 2011-07-09T01:09:06Z 2011-07-09T01:09:06Z Yes I had actually found your question and was trying to relate it to mine by pulling back again via $x \mapsto (x,x)$ or maybe $x \mapsto (x,c)$ but couldn't work it out. Saying that $V$ would &quot;lift&quot; $\kappa \oplus \kappa^{-1}$ was motivated I guess by the thought that a diagram chase might get something like what I'm looking for from the $g-1$ Poincare bundle, which would live on $C \times J^{g-1})$ (maybe) and restrict to some version of $\kappa$ on the Abel image in the fibers over $C$. http://mathoverflow.net/questions/69754/pulling-back-a-line-bundle-on-the-jacobian-to-a-spin-bundle-on-the-curve/69776#69776 Comment by aaron gerding aaron gerding 2011-07-08T22:41:04Z 2011-07-08T22:41:04Z I think this is exactly the point I was missing. Thanks a lot! I guess seeing $\kappa \oplus \kappa^{-1}$ as a pull-back is more subtle, if even possible at all... http://mathoverflow.net/questions/69754/pulling-back-a-line-bundle-on-the-jacobian-to-a-spin-bundle-on-the-curve Comment by aaron gerding aaron gerding 2011-07-08T22:35:59Z 2011-07-08T22:35:59Z Sorry, I just meant I'd like to find a bundle $V$ with $\alpha_c^*(V) \cong \kappa \oplus \kappa^{-1}$...