Given a family of smooth curves over a punctured disk, to extend this family to a family over (a ramified cover of) the whole disk, we have to allow the central fiber to be a stable nodal curve. If we want to extend families of curves together with some “additional data” (e.g. a vector bundle over the family, a fibration in elliptic curves over the family, G-torsors, etc), then the problem becomes more complicated, and in general stable nodal curves are not enough (for example, in the case of curves with a line bundle, or admissible covers). In this talk I will present a result that says that, if we allow the central fiber to be a nodal twisted curve whose coarse space is quasi stable (i.e. it might contain rational components with two marked points), then we can always extend the original family of curves plus “additional data” (e.g. vector bundle, fibration, G-torsor,…) to a family over the whole disk, as long as the “additional data” is given by maps to quotient stacks admitting a projective good moduli space. Moreover, the algorithm for extending this family is explicit.
This is joint work with Andrea Di Lorenzo.