3-folds \(X\subset\mathbb{P}^7\) with a hypersurface of points with \(X\)-rank \(\geq 3\). (English) Zbl 1213.14096
Summary: Let \(X\subset\mathbb P^{2m+1}\), \(m\in\{2,3\}\), be a smooth and non-defective subvariety. Here, we prove that there is a hypersurface of \(\mathbb P^{2m+1}\) formed by points with \(X\)-rank \(\geq 3\) if and only if \(X\) is an OADP (a variety with one apparent double point). All such surfaces and three-folds are classified by C. Ciliberto, M. Mella and F. Russo [J. Algebr. Geom. 13, No. 3, 475–512 (2004; Zbl 1077.14076)].
MSC:
| 14N05 | Projective techniques in algebraic geometry |
| 14J99 | Surfaces and higher-dimensional varieties |
