Abstract: Given a family of smooth curves degenerating to a nodal curve, the canonical series on the smooth members degenerate to a collection of linear series on the nodal curve. Assume the components of the nodal curve meet at points in general position. In the 80’s, Eisenbud and Harris described this collection of linear series when the curve is of compact type. In the early 00’s, Medeiros and I described it when the curve has two components that meet in any number of points. And a few years afterwards, Salehyan and I described it when any two components of the curve meet, a condition that is in some sense the opposite of compact type. For the curves in between these two extremal conditions, there was no general description of the limit collection of linear series. But in the late 10’s, Bainbridge, Chen, Gendron, Grushevsky and Möller found a condition that limits of differentials must satisfy, the so-called global residue condition, in their work on compactifying strata of Abelian differentials. This was the missing piece of the puzzle.
I will talk about work in progress by Amini, Garcez and me, which uses the global residue condition to extend the description mentioned above to all nodal curves.