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

TL;DR

This paper classifies complete $C$-subalgebras of $C[[t]]$ using semigroup invariants, relating them to moduli spaces of curve singularities, and describes their geometric structure as affine varieties.

Contribution

It introduces an explicit algorithm to describe the space of subalgebras with a given semigroup as an affine variety and analyzes its stratification and special cases.

Findings

The space $ R_g$ is an affine variety defined by explicit equations.

Certain semigroups $g$ yield $ R_g$ as an affine space.

Stratification of $ R_g$ by embedding dimension is described.

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.