The Question of Arnold on classification of co-artin subalgebras in singularity theory

TL;DR

This paper classifies co-Artin subalgebras of polynomial algebras containing specific ideals, describing their structure as algebraic varieties, and explicitly characterizing their automorphism groups, advancing the understanding of singularity classification.

Contribution

It provides a detailed classification of certain subalgebras of polynomial rings, including their structure, automorphisms, and explicit algebraic descriptions, up to isomorphism.

Findings

The set of such subalgebras forms a disjoint union of affine algebraic varieties.

Explicit generators and relations are given for the algebras in each variety.

Automorphism groups are explicitly described, with finiteness conditions identified.

Abstract

In \cite[Section 5, p.32]{Arnold-1998}, Arnold writes: "Classification of singularities of curves can be interpreted in dual terms as a description of 'co-artin' subalgebras of finite co-dimension in the algebra of formal series in a single variable (up to isomorphism of the algebra of formal series)." In the paper, such a description is obtained but up to isomorphism of algebraic curves (i.e. this description is finer). Let $K$ be an algebraically closed field of arbitrary characteristic. The aim of the paper is to give a classification (up to isomorphism) of the set of subalgebras $\mathcal{A}$ of the polynomial algebra $K[x]$ that contains the ideal $x^mK[x]$ for some $m\geq 1$. It is proven that the set $\mathcal{A} = \coprod_{m, \Gamma }\mathcal{A} (m, \Gamma )$ is a disjoint union of affine algebraic varieties (where $\Gamma \coprod \{0, m, m+1, \ldots \}$ is the semigroup of the singularity and $m-1$ is the Frobenius number). It is proven that each set $\mathcal{A} (m, \Gamma )$ is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on $\mathcal{A} (m ,\Gamma )$. An isomorphism criterion is given for the algebras in $\mathcal{A}$. For each algebra $A\in \mathcal{A} (m, \Gamma)$, explicit sets of generators and defining relations are given and the automorphism group ${\rm Aut}_K(A)$ is explicitly described. The automorphism group of the algebra $A$ is finite iff the algebra $A$ is not isomorphic to a monomial algebra, and in this case $|{\rm Aut}_K(A)|<{\rm dim}_K(A/\mathfrak{c}_A)$ where $\mathfrak{c}_A$ is the conductor of $A$. The set of orders of the automorphism groups of the algebras in $\mathcal{A} (m , \Gamma )$ is explicitly described.