*(English)*Zbl 1029.32020

Consider the rational mappings $f$ from $\u2102{P}^{1}$ into $\u2102{P}^{2}$ of the form $\left[z\right]\mapsto P\circ {P}_{0}\left(\left[z\right]\right)$, where ${P}_{0}\left([{z}_{0},{z}_{1}]\right)=[{z}_{0}{\overline{z}}_{0},{z}_{0}{\overline{z}}_{1},{z}_{1}{\overline{z}}_{0},{z}_{1}{\overline{z}}_{1}]$ and $P:\u2102{P}^{3}\to \u2102{P}^{2}$ is an onto complex linear map. Two such mappings $f$ are equivalent, if they are equal after biholomorphic transformations of the domain and target, respectively.

The author gives a complete classification for the equivalence problem. For the proof, the author associates to each mapping $f$ a $2\times 2$ complex matrix $K$, and shows that the equivalence of two such mappings implies the similarity of the corresponding $2\times 2$ matrices under matrix transformations satisfying a reality condition. The converse is also true, which is however a consequence of the geometric interpretation of the normal form of the matrix $K$. The complete set of equivalence classes of mappings $f$ consists of a continuous family of embeddings of $\u2102{P}^{1}$ in $\u2102{P}^{2}$ with exactly two elliptic complex tangents, a continuous family of totally real immersions of $\u2102{P}^{1}$ in $\u2102{P}^{2}$ with one point of self-intersection, and finitely many others.

The author also studies the equivalence problem for analogous mappings from $\u2102{P}^{2}$ into $\u2102{P}^{5}$. The complete classification for this case and for the higher dimension case is unsettled in this paper. However, the author gives some interesting examples.