• DOI:10.1007/s00209-018-2225-8
• Corpus ID: 119167868

Slice-polynomial functions and twistor geometry of ruled surfaces in $$\mathbb {CP}^3$$CP3

@article{Altavilla2017SlicepolynomialFA,
  title={Slice-polynomial functions and twistor geometry of ruled surfaces in \$\$\mathbb \{CP\}^3\$\$CP3},
  author={Amedeo Altavilla and Giulia Sarfatti},
  journal={Mathematische Zeitschrift},
  year={2017},
  volume={291},
  pages={1059-1092}
}

• A. Altavilla, G. Sarfatti
• Published 28 December 2017
• Mathematics
• Mathematische Zeitschrift

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$$P^\vee $$P∨ and its extension to the real axis $$P_{\mathbb {R}}$$PR, 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, $$P^\vee $$P∨ and $$P_{\mathbb {R}}$$PR 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 $$\mathcal {C}$$C in $$\mathbb {CP}^3$$CP3 and we conclude by giving some qualitative results on the complex structures induced by $$\mathcal {C}$$C via the twistor projection.