Following the inspiring work of Oda-Seshadri, we define and study a universal stability space V_{g,n} for compactified Jacobians with the property that every element \phi \in V_{g,n} corresponds to a compactification of the universal Jacobian over Mbar_{g,n}, the moduli space of stable n-pointed curves of genus g. The space V_{g,n} comes with a wall and chambers decomposition that corresponds to the elements \phi where the compactification changes. We explicitly describe the stability space and its decomposition. As applications, we show how to resolve the indeterminacy of the Abel-Jacobi morphism and we exhibit non-isomorphic compactified universal Jacobians over Mbar_{g,n} for all (g,n) with g>0 (except for 7 exceptional pairs with low g and n). This is a joint work with Jesse Kass. |

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