Let \(\mathcal{O}\) be a discrete valuation ring. Then \(\mathcal{O}\) is a PID with a unique non-zero prime ideal, and thus is local. Its maximal ideal \(\mathfrak{m}\) is generated by an element \(t\) called a uniformizing parameter and all other ideals have the form \((t^m)\). We can thus define a valuation \(\nu(x)=m\) for \(x\in (t^m)\). We assume that \(\mathcal{O}\) has equi-characteristic 0 which means that both \(\mathcal{O}\) and \(\mathcal{O}/\mathfrak{m}\) have characteristic 0 (this implies, for instance, that \(\mathcal{O}\) contains \(\mathbb{Q}\)). Let \(B\) be an integral domain finitely generated over \(\mathcal{O}\) such that the quotient field of \(\mathcal{O}\) is algebraically closed in the quotient field of \(B\). Assume further that \(B\) is equipped with a non-trivial locally nilpotent \(\mathcal{O}\)-derivation \(\delta\), \(B\) has relative dimension one over \(\mathcal{O}\), and \(B/tB\) is an integral domain. Then \(B\) is called an integral affine \(\mathcal{O}\)-curve. The order of \(B\) is the minimal valuation of all elements in \(\mathcal{O}\cap \delta(B)\).
An example is \(B=\mathcal{O}[x,y]/(t^ny-f(x))\) where \(f\in \mathcal{O}[x]\) is monic. This \(\mathcal{O}\)-curve is said to be of Danielewski type. The author gives intrinsic conditions in terms of order for a general integral affine \(\mathcal{O}\)-curve to be of Danielewski type, but also shows there are examples which are not of this type.
The main theorem in this paper states (in part) that for any integral affine \(\mathcal{O}\)-curve \(B\) with positive order there exists positive integers \(\ell_1,\dots,\ell_{s+1}\) and \(x,y_1,\dots,y_{s+1}\in B\) such that \(B=\mathcal{O}[x,y_1,\dots,y_{s+1}]\) subject to relations \(t^{\ell_{i+1}}y_{i+1}=f_i(x,y_1,\dots,y_i)\) for all \(0\leq i\leq s\) where \(f_i\) are polynomials satisfying certain conditions the author describes.