Wojciech J. Gajda

Abstract:

Every non-singular algebraic curve C has a jacobian J, which is an abelian variety. Choosing a point on the curve determines an abel-jacobi map from C to J. The same constructions can be made in families: given a family of non-singular curves (together with a section), one obtains a family of abelian varieties, and an abel-jacobi map. We are interested in what happens when such a family of non-singular pointed curves degenerates to a singular pointed 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 seemingly-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.