On Severi varieties as intersections of a minimum number of quadrics

Let \({\mathscr{V}}\) be a variety related to the second row of the Freudenthal-Tits Magic square in \(N\)-dimensional projective space over an arbitrary field. We show that there exist \(M\leq N\) quadrics intersecting precisely in \({\mathscr{V}}\) if and only if there exists a subspace of projective dimension \(N-M\) in the secant variety disjoint from the Severi variety. We present some examples of such subspaces of relatively large dimension. In particular, over the real numbers we show that the Cartan variety (related to the exceptional group \({E_6}\)\((\mathbb R)\)) is the set-theoretic intersection of 15 quadrics.