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

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 of the polynomial algebra K [ x ] that contains the ideal x m K [ x ] for some m ≥ 1. It is proven that the set = ∐ m , Γ (m , Γ) is a disjoint union of affine algebraic varieties (where Γ ∐ { 0 , m , m + 1 , ... } is the semigroup of the singularity and m − 1 is the Frobenius number). It is proven that each set (m , Γ) is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on (m , Γ). An isomorphism criterion is given for the algebras in . For each algebra A ∈ (m , Γ) , explicit sets of generators and defining relations are given and the automorphism group 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 | Aut K (A) | < dim K (A / A) where A is the conductor of A. The set of orders of the automorphism groups of the algebras in (m , Γ) is explicitly described. [ABSTRACT FROM AUTHOR]