The versal deformation of elliptic m-fold point curve singularities
Jan Stevens
TL;DR
The paper develops explicit, highly symmetric equations for the versal deformation of the elliptic $m$-fold point $L_{n+1}^n$ (the union of $n+1$ lines in generic position) and reveals an inductive structure in the base space: for $n\geq5$, the base space $B_n$ is isomorphic to the total space of the versal deformation of $L_n^{n-1}$. It leverages Pinkham and Looijenga's framework on deformations with negative weight to construct a fiberwise projectivisation whose projective base $\mathbb{P}(B_n)$ furnishes a modular compactification of the genus-1, $(n+1)$-pointed moduli space, coinciding with Smyth’s compactification $\overline{M}_{1,n+1}(n)$. The authors provide explicit equations for both total and base spaces (including a Pfaffian form in special cases) and establish that $B_n$ is Gorenstein and irreducible with dimension $n+2$; they also discuss the limitations and complexity of extending these computations to general elliptic partition curves and to monomial curves with larger multiplicities, where base spaces can have components of different dimensions. This work connects versal deformation theory, modular compactifications, and the geometry of elliptic singularities in a concrete, computable framework.
Abstract
We give explicit, highly symmetric equations for the versal deformation of the singularity $L_{n+1}^n$ consisting of n+1 lines through the origin in n-dimensional affine space in generic position. These make evident that the base space of the versal deformation of $L_{n+1}^n$ is isomorphic to the total space for $L_{n}^{n-1}$, if n>4. 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 unfeasable 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.