Transverse lines to surfaces over finite fields (Q2663056)

Let \(S\subset \mathbb {P}^3\) be a smooth degree \(d\) surface defined over the finite field \(\mathbb {F}_q\). There are \((S,d,q)\) such that all planes defined over \(\mathbb {F}_q\) are tangent to \(S\). One of the Bertini theorems proved in [\textit{B. Poonen}, Ann. Math. (2) 160, No. 3, 1099--1127 (2005; Zbl 1084.14026)] proves the existence of high degree surfaces transversal to \(S\) and defined over \(\mathbb {F}_q\). The author studies the existence of a line \(L\) defined over \(\mathbb {F}_q\) and transversal to \(S\). The author gives an example with \(d=q+2\) for which it fails. For all \(q\ge d\) the author gives a lower bound for the number of transversal lines defined over \(\mathbb {F}_q\), which implies the existence of at least one such line if \(q\ge d^2\). The main result of this paper gives the existence of a transversal line defined over \(\mathbb {F}_q\) if \(S\) is reflexive and \(q\ge cd\), where \(c =\frac{3+\sqrt{17}}{4}\sim 1.7808\).