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

Abstract

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 \cite{gensalsto} and developed in \cite{AAtwistor}. To any slice-polynomial function $P$ we associate its {\em companion} $P^{\vee }$ and its {\em 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.

30 pages, 5 figures

References in corpus (2)

• Twistor interpretation of slice regular functions
• Twistor lines on algebraic surfaces

Cited by in corpus (6)

• On a class of orientation-preserving maps of ${\mathbb{R}}^4$
• Equivalence of slice semi-regular functions via Sylvester operators
• Twistor geometry of the Flag manifold
• Algebraic surfaces with infinitely many twistor lines
• Three Topological Results on the Twistor Discriminant Locus in the 4-Sphere
• Spherical coefficients of slice regular functions