  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 $x^mK[x]$ 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 $\Gamma \coprod \{0, m, m+1, \ldots \}$ 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 ${\rm Aut}_K(A)$ 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 $|{\rm Aut}_K(A)|<{\rm
dim}_K(A/\mathfrak{c}_A)$ 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.
