Projective techniques in twistor geometry

Abstract This survey explores the interplay between twistor geometry and projective geometry, focusing on their applications to algebraic surfaces. We explore two main topics: the inclusion of twistor fibers and lines in these surfaces, and the behavior of twistor discriminant loci, with a particular focus on degree-2 surfaces. The study highlights contributions from Ballico and collaborators, comparing the twistor spaces of $${\mathbb{C}\mathbb{P}}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> and the flag threefold $${\mathbb {F}},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> which reveal fascinating contrasts and parallels. Key findings include a detailed analysis of surfaces in $${\mathbb{C}\mathbb{P}}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> and $${\mathbb {F}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> that either contain finite or infinite twistor fibers. The survey also touches on cubic surfaces in $${\mathbb{C}\mathbb{P}}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> and their counterparts in $${\mathbb {F}},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where the configurations of twistor fibers lead to intriguing results. Special attention is given to surfaces of twistor degree 2, including how their geometry and singularities interact with twistor projections. In particular, we discuss smooth and singular surfaces in $${\mathbb {F}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> of bidegree (1, 1) and (0, 2), as well as their discriminant loci.