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

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 $x^{m}K[x]$ for some $m\ge 1$. It is proven that the set $\mathcal{A}={\coprod }_{m,\mathrm{\Gamma }}\mathcal{A}(m,\mathrm{\Gamma })$ is a disjoint union of affine algebraic varieties (where $\mathrm{\Gamma }\coprod \{0,m,m+1,\dots \}$ is the semigroup of the singularity and $m-1$ is the Frobenius number). It is proven that each set $\mathcal{A}(m,\mathrm{\Gamma })$ is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on $\mathcal{A}(m,\mathrm{\Gamma })$. An isomorphism criterion is given for the algebras in $\mathcal{A}$. For each algebra $A\in \mathcal{A}(m,\mathrm{\Gamma })$, explicit sets of generators and defining relations are given and the automorphism group ${\mathrm{A}\mathrm{u}\mathrm{t}}_K(A)$ 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 $\mathrm{\mid }{\mathrm{A}\mathrm{u}\mathrm{t}}_K(A)\mathrm{\mid }<{\mathrm{d}\mathrm{i}\mathrm{m}}_K(A\mathrm{/}{\mathfrak{c}}_A)$ where ${\mathfrak{c}}_A$ is the conductor of $A$. The set of orders of the automorphism groups of the algebras in $\mathcal{A}(m,\mathrm{\Gamma })$ is explicitly described.