The versal deformation of elliptic m-fold point curve singularities

Journal article, 2025

We give explicit, highly symmetric equations for the versal deformation of the singularity Ln+1n consisting of n+1 lines through the origin in An(k) in generic position. These make evident that the base space of the versal deformation of Ln+1n is isomorphic to the total space for Lnn-1, if n≥5. By induction it follows that the base space is irreducible and Gorenstein. We discuss the known connection to a modular compactification of the moduli space of (n+1)-pointed curves of genus 1. For other elliptic partition curves it seems unfeasible to compute the versal deformation in general. It is doubtful whether the base space is Gorenstein. For rational partition curves we show that the base space in general has components of different dimensions.