## Abstract

We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration \({\pi : \mathbb{CP}^3 \rightarrow S^4}\). 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 special 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.

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GNSAGA of INdAM

MIUR PRIN 2015 “Geometria delle varietà algebriche”

SIR grant “NEWHOLITE – New methods in holomorphic iteration” n. RBSI14CFME, SIR grant AnHyC – Analytic aspects in complex and hypercomplex geometry n. RBSI14DYEB and Fondazione Bruno Kessler–CIRM (Trento) “Research In Pairs” program. Moreover, the present paper was submitted while the first author was a postdoc (assegno di ricerca) at Dipartimento Di Matematica, Università di Roma “Tor Vergata”.

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Altavilla, A., Ballico, E. Three Topological Results on the Twistor Discriminant Locus in the 4-Sphere.
*Milan J. Math.* **87**, 57–72 (2019). https://doi.org/10.1007/s00032-019-00292-5

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DOI: https://doi.org/10.1007/s00032-019-00292-5