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

Abstract

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 [Formula: see text] 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 [Formula: see text] of the polynomial algebra [Formula: see text] that contains the ideal [Formula: see text] for some [Formula: see text]. It is proven that the set [Formula: see text] is a disjoint union of affine algebraic varieties (where [Formula: see text] is the semigroup of the singularity and [Formula: see text] is the Frobenius number). It is proven that each set [Formula: see text] is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on [Formula: see text]. An isomorphism criterion is given for the algebras in [Formula: see text]. For each algebra [Formula: see text], explicit sets of generators and defining relations are given and the automorphism group [Formula: see text] is explicitly described. The automorphism group of the algebra [Formula: see text] is finite if and only if the algebra [Formula: see text] is not isomorphic to a monomial algebra, and in this case [Formula: see text] where [Formula: see text] is the conductor of [Formula: see text]. The set of orders of the automorphism groups of the algebras in [Formula: see text] is explicitly described.