2022/06/16 by V. V. Bavula, Bavula, V. V.
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2206.08286
openalex publication_date 2022/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 A of the polynomial algebra K[x] that contains the ideal xmK[x] for some m≥ 1. It is proven that the set A = \coprodm, ΓA (m, Γ) is a disjoint union of affine algebraic varieties (where Γ\coprod \0, m, m+1, … \ is the semigroup of the singularity and m-1 is the Frobenius number). It is proven that each set A (m, Γ) is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on A (m ,Γ). An isomorphism criterion is given for the algebras in A. For each algebra A∈ A (m, Γ), explicit sets of generators and defining relations are given and the automorphism group \rm AutK(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 AutK(A)|<\rm dimK(A/\mathfrakcA) where \mathfrakcA is the conductor of A. The set of orders of the automorphism groups of the algebras in A (m , Γ) is explicitly described.