Surfaces in the flag threefold containing smooth conics and twistor fibers

TL;DR

This paper investigates smooth conics and twistor fibers in the flag threefold, establishing bounds on their occurrence in surfaces and constructing examples of surfaces with infinitely many twistor fibers, revealing geometric constraints.

Contribution

It provides bounds on the number of smooth conics in surfaces and constructs new surfaces with infinitely many twistor fibers, advancing understanding of their geometric configurations.

Findings

Maximum smooth conics in a surface are bounded.

Surfaces of bidegree (1,1) contain infinitely many twistor fibers.

Construction method for surfaces of bidegree (a,a) with infinitely many twistor fibers.

Abstract

We study smooth integral curves of bidegree $(1,1)$, called \textit{smooth conics}, in the flag threefold $\mathbb{F}$. The study is motivated by the fact that the family of smooth conics contains the set of fibers of the twistor projection $\mathbb{F}\to\mathbb{CP}^{2}$. We give a bound on the maximum number of smooth conics contained in a smooth surface $S\subset\mathbb{F}$. Then, we show qualitative properties of algebraic surfaces containing a prescribed number of smooth conics. Lastly, we study surfaces containing infinitely many twistor fibers. We show that the only smooth cases are surfaces of bidegree $(1,1)$. Then, for any integer $a>1$, we exhibit a method to construct an integral surface of bidegree $(a,a)$ containing infinitely many twistor fibers.