David Holmes: A Néron model of the universal jacobian

Every non-singular algebraic curve C has a jacobian J, which is a smooth projective group variety. Given a family of non-singular curves one can construct a family of jacobians. We are interested in what happens to the family of jacobians when the family of non-singular curves degenerates to a singular curve. In the case where the base-space of the family has dimension 1 (a `1-parameter family’), this is completely understood due to work of André Néron in the ‘60s. However, when the base space has higher dimension things become more difficult. We describe a new combinatorial invariant which controls these degenerations. In the case of the jacobian of the universal stable curve, we will use this to construct a `minimal’ base-change after which a Néron model exists. If time allows, we will discuss connections to number theory and Gromov–Witten theory.