Threefolds of the Klein quadric and trisecant lines (Q1841855)

Let \(F\) denote an algebraically closed field with \(\text{char}(F)=p >0\), let \({\mathcal K} \subseteq {\text{PG}}(5,F)\) be the Klein quadric and let \(U \subseteq {\mathcal K}\) be an irreducible quasi projective variety with \(\dim(U) = 3\). A line \(D\) is said to be a true trisecant of \(U\) if \(|(U \cap D)|\geq 3\). A line \(D\) is said to be a generalized trisecant of \(U\) if either \(D\) is a true trisecant line to \(U\) or \(D \cap U\) is an effective Cartier divisor of \(D\) with degree at least 3. The authors prove that every point of \(U\) is contained in a true secant line \(D\) to \(U\) and a generalized trisecant line to the Zariski closure of \(U\).