A Bertini type theorem for pencils over finite fields (Q2667079)

Let \(k\) be a finite field and \(\overline{k}\) its algebraic closure. The aim of this paper is to raise the Bertini's type problem for a pencil of hypersurfaces over \(k\). Let \(X_1,X_2\subset \mathbb{P}^n\), \(n\ge 2\), be degree \(d\ge 2\) hypersurfaces defined over \(k\). This pencils is said to have property (T) if each member of the pencil has a transversal hyperplane over \(\overline{k}\) and the pencil has a smooth member over \(\overline{k}\). Let \(p\) be the characteristic of \(k\). The author proves that if \(p\nmid n(d-1)\) there is an extension \(k'\) of \(k\) such that for every finite extension \(k''\) of \(k'\) there is a hyperplane \(H\) defined over \(k''\) and transversal to all elements of the pencil \(sX_1+tX_2\), \(s, t\in k''\). Moreover, the degree of the extension \(k'/k\) only depends on \(n\) and \(d\). The author gives several examples explaining the assumptions of the theorem.