On coverings of stable curves

I will report on joint work with Dan Abramovich and Alessio Corti.

Compact Riemann surfaces, or smooth algebraic curves, are basic objects in geometry. When studying families of smooth algebraic curves of fixed genus one has to deal with the problem that these families do not form a compact space; in other words, the limit of a family of smooth curves is not necessarily smooth. It was shown by Deligne and Mumford that we can get a compact space by introducing some singular curves, called stable curves; these have the mildest possible type of singularities, namely nodes.

One often wants to study Riemann surfaces with some additional structure: for example, one might be interested in Riemann surfaces with a specified topological cover, or a fixed element in the first homology group. But then stable curves don't do the job, because they do not have enough covers, and their first homology group is too small.

It was discovered by Dan Abramovich and myself that in many cases the natural limits are not given by bare stable curves, but by what we call twisted curves. Very roughly, the idea is that one has to substitute each singular point with the classifying space of a cyclic group.

I will start by reviewing the classical theory of stable curves and explain its inadequacies, and conclude by giving some idea of what twisted curves are, and why they give a good theory.