Differentials and Singularities

Given a holomorphic differential on a smooth algebraic curve, we associate to it a Gorenstein curve singularity with Gm-action.  Conversely, we show that every isolated Gorenstein curve singularity with Gm-action appears in this way.  This construction reveals a fascinating relation between differentials and singularities, where the zero orders of the differentials determine the combinatorial data of the singularities.  In this talk, I’ll provide many concrete examples of such singularities, and explain how the study of deformations of these singularities can help us better understand the geometry of moduli spaces of differentials.  This is based on joint work with Fei Yu (Zhejiang University).