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

TL;DR

This paper investigates the topology of the twistor discriminant locus for algebraic surfaces in the 4-sphere, revealing its dimension, intersection configurations, and a decomposition related to singularities and dual varieties.

Contribution

It provides three new topological results on the twistor discriminant locus, including its dimension, intersection configurations, and a decomposition related to singularities and dual varieties.

Findings

Discovered that the discriminant locus generally has real dimension 2.

Identified four possible intersection configurations with twistor lines.

Decomposed the discriminant locus of cones into singular and dual parts.

Abstract

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.