Slice-polynomial functions and twistor geometry of ruled surfaces in $$mathbb {CP}^3$$ CP 3

摘要

In the present paper we introduce the class of slice-polynomial functions:slice regular functions {defined over the quaternions, outside the real axis,}whose restriction to any complex half-plane is a polynomial. These functionsnaturally emerge in the twistor interpretation of slice regularity introducedin cite{gensalsto} and developed in cite{AAtwistor}. To any slice-polynomialfunction $P$ we associate its {em companion} $P^ee$ and its {em extension}to the real axis $P_mathbb{R}$, that are quaternionic functions naturallyrelated to $P$. Then, using the theory of twistor spaces, we are able to showthat for any quaternion $q$ the {cardinality of simultaneous} pre-images of $q$via $P$, $P^ee$ and $P_mathbb{R}$ is generically constant, giving a notionof degree. With the brand new tool of slice-polynomial functions, we {compute}the twistor discriminant locus of a cubic scroll $mathcal{C}$ in$mathbb{CP}^3$ and we conclude by giving some qualitative results on thecomplex structures induced by $mathcal{C}$ via the twistor projection.