Twistor geometry of the flag manifold

A study is made of algebraic curves and surfaces in the flag manifold F = SU(3)/T2, and their configuration relative to the twistor projection π from F to the complex projective plane P2, defined with the help of an anti-holomorphic involution j. This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space P3 of S4. Deformations of twistor fibres project to real surfaces in P2, whose metric geometry is investigated. Attention is then focussed on toric Del Pezzo surfaces that are the simplest type of surfaces in F of bidegree (1, 1). These surfaces define orthogonal complex structures on specified dense open subsets of P2 relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree (1,1) are determined, and bounds given on the number of twistor fibres that are contained in more general algebraic surfaces in F.