# Three Topological Results on the Twistor Discriminant Locus in the 4-Sphere

created by altavilla on 24 Aug 2018
modified on 11 Mar 2019

[BibTeX]

published

Inserted: 24 aug 2018
Last Updated: 11 mar 2019

Journal: Milan Journal of Mathematics
Pages: 16
Year: 2019
Doi: https://doi.org/10.1007/s00032-019-00292-5

ArXiv: 1808.07806 PDF
We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration $\pi:\mathbb{CP}^{3}\to S^{4}$. We prove three results about the topology of the twistor discriminant locus of an algebraic surface in $\mathbb{CP}^{3}$. First of all we prove that, with the exception of two exceptional cases, the real dimension of the twistor discriminant locus of an algebraic surface is always equal to 2. Secondly we describe the possible intersections of a general surface with the family of twistor lines: we find that only 4 configurations are possible and for each of them we compute the dimension. Lastly we give a decomposition of the twistor discriminant locus of a given cone in terms of its singular locus and its dual variety.