Slice-Polynomial Functions and Twistor Geometry of Ruled Surfaces in $\mathbb{CP}^3$

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 functions naturally emerge in the twistor interpretation of slice regularity introduced in Gentili et al. (J Eur Math Soc 16(11):2323–2353, 2014) and developed in Altavilla (J Geom Phys 123:184–208, 2018). To any slice-polynomial function P we associate its companion\(P^\vee \) and its extension to the real axis \(P_{\mathbb {R}}\), that are quaternionic functions naturally related to P. Then, using the theory of twistor spaces, we are able to show that for any quaternion q the cardinality of simultaneous pre-images of q via P, \(P^\vee \) and \(P_{\mathbb {R}}\) is generically constant, giving a notion of 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 the complex structures induced by \(\mathcal {C}\) via the twistor projection.