Three-dimensional scrolls in \(\mathbb{P}^ 6\) (Q1924568)

If a \(n\)-dimensional scroll \(X(=\) a \(\mathbb{P}^{n-1}\)-bundle) over the smooth curve \(C\) of genus \(g\) is embedded in \(\mathbb{P}^N\) by linearly embedded fibers then \(N\geq 2n-1\); and if \(N=2 n-1\) then, by the Barth-Lefschetz theorem, \(g=0\). If \(N=2n\), the only known examples are scrolls over curves with \(g=0,1\) and the problem is to prove this or to find a scroll with \(N=2n\) and \(g\geq 2\). For \(n=2\) the problem has been solved affirmatively by \textit{Alf Aure}. In this paper, following the idea of Aure, the author gives an affirmative answer also for \(n=3\). The proof is based on the study of the family of 3-planes \(\mathbb{P}_i\) in \(\mathbb{P}^6\) intersecting all the fibers of the scroll.