Surfaces in the Flag Threefold Containing Smooth Conics and Twistor Fibers

Abstract

We study smooth integral curves of bidegree (1, 1), called 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 \(\mathbb {F}\rightarrow {\mathbb {C}}{\mathbb {P}}^{2}\). 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. Last, 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.