Classifying complete $\mathbb{C}$-subalgebras of $\mathbb{C}[[t]]$

Abstract

We address the problem of classifying complete $\mathbb{C}$-subalgebras of $\mathbb{C}[[t]]$. A discrete invariant for this classification problem is the semigroup of orders of the elements in a given $\mathbb{C}$-subalgebra. Hence we can define the space $\mathcal{R}_{\Gamma}$ of all $\mathbb{C}$-subalgebras of $\mathbb{C}[[t]]$ with semigroup $\Gamma$. After relating this space to the Zariski moduli space of curve singularities and to a moduli space of global singular curves, we prove that $\mathcal{R}_{\Gamma}$ is an affine variety by describing its defining equations in an ambient affine space in terms of an explicit algorithm. Moreover, we identify certain types of semigroups $\Gamma$ for which $\mathcal{R}_{\Gamma}$ is always an affine space, and for general $\Gamma$ we describe the stratification of $\mathcal{R}_{\Gamma}$ by embedding dimension. We also describe the natural map from $\mathcal{R}_{\Gamma}$ to the Zariski moduli space in some special cases. Explicit examples are provided throughout.