Kodaira dimension of embeddings of the line in the plane (Q1078268)

For a divisor V of a function field L in two variables over an algebraically closed field k, one considers a complete non-singular model X, such that the center of V is a non-singular curve D. The logarithmic Kodaira dimension \(\kappa\) (X-D) is independent of the choice of X and is denoted by \(\kappa\) (V). As well known, over a field k of characteristic p\(>0\), there are non trivial closed embeddings \(i:\quad {\mathbb{A}}^ 1\to {\mathbb{A}}^ 2.\) Denoting by f the defining equation of \(i({\mathbb{A}}^ 1)\), the author gives a coarse classification of those embeddings in terms of \(\kappa\) (f). Especially he proves, that \(\kappa (f)=-\infty\) if and only if \(i({\mathbb{A}}^ 1)\) is a coordinate line, i.e.: if there exists a function g with \(k[x,y]=k[f,g]\).