Consider a stable curve X of genus g lying on the boundary of the Deligne-Mumford compactification of the moduli space of curves. I will talk about joint work with Esteves, and with Esteves—Garcez, in which we study the following problems:

– Given a family of smooth projective curves of genus g degenerating to X, what is the limit of the corresponding g-dimensional spaces of Abelian differentials a.k.a canonical series?

– How to construct a parameter space for all the limits, along any possible degenerating family?

We answer the first by associating a polyhedral tiling of a simplex to the family, with tiles decorated by g dimensional spaces of meromorphic differentials on X, that encode the limit.
We answer the second by constructing a projective variety of limit canonical series associated to X, under the assumption that the nodes of X are in general position on each component of X. This extends to any topology the previous work on the problem done by Eisenbud—Harris 87 (curves of compact type) and by Esteves-Medeiros 2002 (curves with two components). The result is effective and provides an explicit description of the limits, as well as a stratification of the variety of limit canonical series by a rational fan defined on the space of edge lengths of the dual graph of X.

A key role in our approach is played by combinatorial strucures called polymatroids and a discrete notion of convexity for functions defined on power sets called submodularity.