Resumen de On Picard bundles over Prym varieties

L. Brambila-Paz, Esteban Gómez González , F. Pioli

• Let $\pi: Y\rightarrow X$ be a covering between non-singular irreducible projective curves. The Jacobian $J(Y )$ has two natural subvarieties, namely, the Prym variety $P$ and the variety $\pi^\ast(J(X))$. We prove that the restriction of the Picard bundle to the subvariety $\pi^\ast(J(X))$ is stable. Moreover, if $\widetilde P$ is a principally polarized Prym- Tyurin variety associated with $P$, we prove that the induced Abel-Prym morphism $\widetilde p: Y\rightarrow\widetilde P$ is birational to its image for genus $g_X > 2$ and deg $\pi\not= 2$. We use this result to prove that the Picard bundle over the Prym variety is simple and moreover is stable when $\widetilde p$ is not birational onto its image.