Summary: In [V. I. Arnold, Simple singularities of curves, Proc. Steklov Inst. Math. 226(3) (1999) 20–28, Sec. 5, p. 32], 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 \(m \geq 1\). It is proven that the set \(\mathcal{A}= \coprod_{m, \Gamma} \mathcal{A}(m,\Gamma)\) is a disjoint union of affine algebraic varieties (where \(\Gamma\coprod\{0, m, m+1, \dots\}\) 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 \(\operatorname{Aut}_K(A)\) is explicitly described. The automorphism group of the algebra \(A\) is finite if and only if the algebra \(A\) is not isomorphic to a monomial algebra, and in this case \(|\operatorname{Aut}_K(A)| < \dim_K(A/\mathfrak{c}_A)\) 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.