Twistor fibers in hypersurfaces of the flag threefold

TL;DR

This paper investigates surfaces within the flag threefold that contain twistor fibers, establishing conditions for their existence and non-existence, and providing explicit constructions for certain cases.

Contribution

It proves the non-existence of certain irreducible surfaces containing many twistor fibers and constructs surfaces containing specified fibers, advancing understanding of twistor fibers in flag threefolds.

Findings

No irreducible surface of bidegree (1,d) contains d+2 twistor fibers in general position.

Existence of surfaces of bidegree (1,d) containing any (d+1) twistor fibers satisfying mild constraints.

Results are refined for d=2 and d=3, removing generality assumptions.

Abstract

We study surfaces of bidegree (1,d) contained in the flag threefold in relation to the twistor projection. In particular, we focus on the number and the arrangement of twistor fibers contained in such surfaces. First, we prove that there is no irreducible surface of bidegree (1,d) containing d+2 twistor fibers in general position. On the other hand, given any collection of (d+1) twistor fibers satisfying a mild natural constraint, we prove the existence of a surface of bidegree (1,d) that contains them. We improve our results for d=2 or d=3, by removing all the generality hypotheses.