Surfaces in the flag threefold containing smooth conics and twistor fibers

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 $\mathbb{F}\to {\mathbb{C}\mathbb{P}}^2$. 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.